diff --git a/doc/source/index.rst b/doc/source/index.rst
index b442d44cc..e18c12914 100644
--- a/doc/source/index.rst
+++ b/doc/source/index.rst
@@ -36,6 +36,7 @@ EKO is ...
     theory/FlavorSpace
     theory/pQCD
     theory/DGLAP
+    theory/N3LO_ad
     theory/Matching
 
     zzz-refs
diff --git a/doc/source/refs.bib b/doc/source/refs.bib
index 8e7218349..06c1804f8 100644
--- a/doc/source/refs.bib
+++ b/doc/source/refs.bib
@@ -654,3 +654,56 @@ @article{Luthe:2017ttg
     pages = "166",
     year = "2017"
 }
+
+@article{Davies:2016jie,
+    author = "Davies, J. and Vogt, A. and Ruijl, B. and Ueda, T. and Vermaseren, J. A. M.",
+    title = "{Large-$n_f$ contributions to the four-loop splitting functions in QCD}",
+    eprint = "1610.07477",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "LTH-1106, NIKHEF-2016-049",
+    doi = "10.1016/j.nuclphysb.2016.12.012",
+    journal = "Nucl. Phys. B",
+    volume = "915",
+    pages = "335--362",
+    year = "2017"
+}
+
+@article{Moch:2017uml,
+    author = "Moch, S. and Ruijl, B. and Ueda, T. and Vermaseren, J. A. M. and Vogt, A.",
+    title = "{Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond}",
+    eprint = "1707.08315",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "DESY-17-106, NIKHEF-2017-034, LTH-1139",
+    doi = "10.1007/JHEP10(2017)041",
+    journal = "JHEP",
+    volume = "10",
+    pages = "041",
+    year = "2017"
+}
+
+@article{Davies:2022ofz,
+    author = "Davies, J. and Kom, C. -H. and Moch, S. and Vogt, A.",
+    title = "{Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering}",
+    eprint = "2202.10362",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-ph",
+    reportNumber = "LTH 1289, DESY 21-225",
+    month = "2",
+    year = "2022"
+}
+
+@article{Henn:2019swt,
+    author = "Henn, Johannes M. and Korchemsky, Gregory P. and Mistlberger, Bernhard",
+    title = "{The full four-loop cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills and QCD}",
+    eprint = "1911.10174",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-th",
+    reportNumber = "IPhT-T19/155, MIT-CTP/5160, MPP-2019-233",
+    doi = "10.1007/JHEP04(2020)018",
+    journal = "JHEP",
+    volume = "04",
+    pages = "018",
+    year = "2020"
+}
diff --git a/doc/source/theory/N3LO_ad.rst b/doc/source/theory/N3LO_ad.rst
new file mode 100644
index 000000000..38bd1af46
--- /dev/null
+++ b/doc/source/theory/N3LO_ad.rst
@@ -0,0 +1,144 @@
+N3LO Anomalous Dimensions
+=========================
+
+The |N3LO| |QCD| anomalous dimensions :math:`\gamma^{(3)}` are not yet fully known,
+since they rely on the calculation of 4-loop DIS integrals.
+Moreover the analytical structure of these function is already known to be complicated
+since in Mellin space it will included harmonics sum up to weight 7, for which an
+analytical contribution is not available.
+
+Here we describe the various assumptions and limits used in order to reconstruct a parametrization
+that can approximate their contribution.
+In particular we will take advantage of some known physical constrain,
+such as large-x limit, small-x limit, and sum rules, in order to make our reconstruction reasonable.
+
+Generally we remark that the large-x limit correspond to large-N in Mellin space
+where the leading contribution comes from the harmonics :math:`S_1(N)`,
+while the small-x region corresponds to poles at :math:`N=0,1` depending on the type of
+divergence.
+
+In any case |N3LO| |DGLAP| evolution at small-x, especially for singlet-like PDFs, will not be reliable
+until the splitting function resummation will not be available up to NNLL.
+
+Non-singlet sector
+------------------
+
+In the non-singlet sector we construct a parametrization for
+:math:`\gamma_{ns,-}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,s}^{(3)}` where:
+
+    .. math ::
+        \gamma_{ns,s}^{(3)} = \gamma_{ns,v}^{(3)} - \gamma_{ns,-}^{(3)}
+
+In particular, making explicitly the dependence on :math:`n_f`, the non-singlet anomalous dimensions include
+the following terms:
+
+    .. list-table:: non-singlet 4-loop Anomalous Dimensions
+        :header-rows: 1
+
+        *   -
+            - :math:`n_{f}^0`
+            - :math:`n_{f}^1`
+            - :math:`n_{f}^2`
+            - :math:`n_{f}^3`
+
+        *   - :math:`\gamma_{ns,-}^{(3)}`
+            - |T|
+            - |T|
+            - |T|
+            - |T|
+
+        *   - :math:`\gamma_{ns,+}^{(3)}`
+            - |T|
+            - |T|
+            - |T|
+            - |T|
+
+        *   - :math:`\gamma_{ns,s}^{(3)}`
+            -
+            - |T|
+            - |T|
+            -
+
+Some of these parts are known analytically exactly (:math:`\propto n_f^2,n_f^3`),
+while others are available only in the large :math`N_c` limit (:math:`\propto n_f^0,n_f^1`).
+In |EKO| they are implemented as follows:
+
+    * the part proportional to :math:`n_f^3` is common for :math:`\gamma_{ns,+}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,v}^{(3)}`
+      and is exact :cite:`Davies:2016jie` (Eq. 3.6).
+
+    * In :math:`\gamma_{ns,s}^{(3)}` the part proportional to :math:`n_f^2`
+      is exact :cite:`Davies:2016jie` (Eq. 3.5).
+
+    * In :math:`\gamma_{ns,s}^{(3)}` the part proportional to :math:`n_f^1` is
+      parametrized in x-space and copied from :cite:`Moch:2017uml` (Eq. 4.19, 4.20).
+
+    * The remaining contributions include the following constrains:
+
+        -   The small-x limit, given in the large :math:`N_c` approximation by
+            :cite:`Davies:2022ofz` (see Eq. 3.3, 3.8, 3.9, 3.10) and coming
+            from small-x resummation.
+            This part contains the so called double logarithms:
+
+            .. math ::
+                \ln(x)^k \quad k=1,..,6, \quad \mathcal{M}[\ln^k(x)] = \frac{1}{N^{(k+1)}}
+
+            Note the expressions are evaluated with the exact values of the |QCD|
+            Casimir invariants, to better agree with the :cite:`Moch:2017uml` parametrization.
+
+        -   The large-N limit :cite:`Moch:2017uml`, which reads (Eq. 2.17):
+
+            .. math ::
+                \gamma_{ns} \approx A_4 S_1(N) - B_4 + C_4 \frac{S_1(N)}{N} - (D_4 + \frac{1}{2} A_4) \frac{1}{N} + \mathcal{O}(\frac{\ln^k(N)}{N^2})
+
+            This limit is common for all :math:`\gamma_{ns,+}^{(3)},\gamma_{ns,-}^{(3)},\gamma_{ns,v}^{(3)}`.
+            The coefficient :math:`A_4`, being related to the twist-2 spin-N operators,
+            can be obtained from the |QCD| cusp calculation
+            :cite:`Henn:2019swt`, while the :math:`B_4` is fixed by the integral of the 4-loop splitting function.
+            :math:`C_4,D_4` instead can be computed directly from lower order splitting functions.
+            From large-x resummation :cite:`Davies:2016jie`, it is possible to infer further constrains
+            on sub-leading terms :math:`\frac{\ln(N)^k}{N^2}`, since the non-singlet splitting
+            functions contain only terms :math:`(1-x)^a\ln^k(1-x)` with :math:`a \ge 1`.
+
+        -   The 8 lowest odd or even N moments provided in :cite:`Moch:2017uml`, where
+            from quark number conservation we can trivially obtain:
+            :math:`\gamma_{ns,s}(1)=\gamma_{ns,-}(1)=0`.
+
+        -   The difference between the known moments and the known limits is parametrized
+            in Mellin space. The basis includes:
+
+            .. list-table::
+                :header-rows: 1
+
+                *   - x-space
+                    - N-space
+                *   - :math:`\delta(1-x)`
+                    - 1
+                *   - :math:`(1-x)\ln(1-x)`
+                    - :math:`\mathcal{M}[(1-x)\ln(1-x)] \approx \frac{S_1(N)}{N^2}`
+                *   - :math:`(1-x)\ln^2(1-x)`
+                    - :math:`\mathcal{M}[(1-x)\ln^2(1-x)] \approx \frac{S_1^2(N)}{N^2}`
+                *   - :math:`(1-x)\ln^3(1-x)`
+                    - :math:`\mathcal{M}[(1-x)\ln^3(1-x)] \approx \frac{S_1^3(N)}{N^2}`
+                *   - :math:`- Li_2(x) + \zeta_2`
+                    - :math:`\frac{S_1(N)}{N^2}`
+
+            which model the sub-leading differences in the :math:`N\to \infty` limit, and:
+
+            .. list-table::
+                :header-rows: 1
+
+                *   - x-space
+                    - N-space
+                *   - :math:`x\ln(x)`
+                    - :math:`\frac{1}{(N+1)^2}`
+                *   - :math:`\frac{x}{2}\ln^2(x)`
+                    - :math:`\frac{1}{(N+1)^3}`
+
+
+            to help the convergence in the small-N region. Finally we add a polynomial part
+            :math:`x^{2(3)}` which corresponds to simple poles at :math:`N=-2,-3`
+            respectively for :math:`\gamma_{ns,+},\gamma_{ns,-}`.
+
+            Note that the constant coefficient is included in the fit, following the procedure done
+            in :cite:`Moch:2017uml` (section 4), to achieve a better accuracy.
+            It is checked that this contribution is much more smaller than the values of :math:`B_4`.
diff --git a/extras/n3lo_ad.tar.gz b/extras/n3lo_ad.tar.gz
new file mode 100644
index 000000000..7b0cbe595
Binary files /dev/null and b/extras/n3lo_ad.tar.gz differ
diff --git a/poetry.lock b/poetry.lock
index e107b38c6..aa4c6da1f 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -83,7 +83,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 name = "attrs"
 version = "21.4.0"
 description = "Classes Without Boilerplate"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
@@ -139,7 +139,7 @@ docs = ["Sphinx (>=4.3.2,<5.0.0)", "sphinx-rtd-theme (>=1.0.0,<2.0.0)", "sphinxc
 name = "beautifulsoup4"
 version = "4.11.1"
 description = "Screen-scraping library"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6.0"
 
@@ -154,7 +154,7 @@ lxml = ["lxml"]
 name = "bleach"
 version = "5.0.0"
 description = "An easy safelist-based HTML-sanitizing tool."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -168,17 +168,17 @@ dev = ["pip-tools (==6.5.1)", "pytest (==7.1.1)", "flake8 (==4.0.1)", "tox (==3.
 
 [[package]]
 name = "certifi"
-version = "2021.10.8"
+version = "2022.5.18.1"
 description = "Python package for providing Mozilla's CA Bundle."
 category = "main"
 optional = false
-python-versions = "*"
+python-versions = ">=3.6"
 
 [[package]]
 name = "cffi"
 version = "1.15.0"
 description = "Foreign Function Interface for Python calling C code."
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -236,14 +236,14 @@ test = ["flake8 (==3.7.8)", "hypothesis (==3.55.3)"]
 
 [[package]]
 name = "coverage"
-version = "6.3.3"
+version = "6.4.1"
 description = "Code coverage measurement for Python"
 category = "dev"
 optional = false
 python-versions = ">=3.7"
 
 [package.dependencies]
-tomli = {version = "*", optional = true, markers = "extra == \"toml\""}
+tomli = {version = "*", optional = true, markers = "python_full_version <= \"3.11.0a6\" and extra == \"toml\""}
 
 [package.extras]
 toml = ["tomli"]
@@ -276,17 +276,17 @@ python-versions = ">=3.5"
 name = "defusedxml"
 version = "0.7.1"
 description = "XML bomb protection for Python stdlib modules"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
 [[package]]
 name = "dill"
-version = "0.3.4"
+version = "0.3.5.1"
 description = "serialize all of python"
 category = "dev"
 optional = false
-python-versions = ">=2.7, !=3.0.*"
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*, !=3.5.*, !=3.6.*"
 
 [package.extras]
 graph = ["objgraph (>=1.7.2)"]
@@ -311,7 +311,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 name = "entrypoints"
 version = "0.4"
 description = "Discover and load entry points from installed packages."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -339,7 +339,7 @@ pyrepl = ">=0.8.2"
 name = "fastjsonschema"
 version = "2.15.3"
 description = "Fastest Python implementation of JSON schema"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -348,7 +348,7 @@ devel = ["colorama", "jsonschema", "json-spec", "pylint", "pytest", "pytest-benc
 
 [[package]]
 name = "filelock"
-version = "3.6.0"
+version = "3.7.1"
 description = "A platform independent file lock."
 category = "dev"
 optional = false
@@ -393,7 +393,7 @@ docs = ["sphinx"]
 
 [[package]]
 name = "identify"
-version = "2.5.0"
+version = "2.5.1"
 description = "File identification library for Python"
 category = "dev"
 optional = false
@@ -420,7 +420,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
 [[package]]
 name = "importlib-metadata"
-version = "4.11.3"
+version = "4.11.4"
 description = "Read metadata from Python packages"
 category = "main"
 optional = false
@@ -438,7 +438,7 @@ testing = ["pytest (>=6)", "pytest-checkdocs (>=2.4)", "pytest-flake8", "pytest-
 name = "importlib-resources"
 version = "5.7.1"
 description = "Read resources from Python packages"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -482,7 +482,7 @@ test = ["pytest (>=6.0)", "pytest-cov", "flaky", "ipyparallel", "pre-commit", "p
 
 [[package]]
 name = "ipython"
-version = "8.3.0"
+version = "8.4.0"
 description = "IPython: Productive Interactive Computing"
 category = "main"
 optional = false
@@ -560,9 +560,9 @@ i18n = ["Babel (>=2.7)"]
 
 [[package]]
 name = "jsonschema"
-version = "4.5.1"
+version = "4.6.0"
 description = "An implementation of JSON Schema validation for Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -573,13 +573,13 @@ pyrsistent = ">=0.14.0,<0.17.0 || >0.17.0,<0.17.1 || >0.17.1,<0.17.2 || >0.17.2"
 
 [package.extras]
 format = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3987", "uri-template", "webcolors (>=1.11)"]
-format_nongpl = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3986-validator (>0.1.0)", "uri-template", "webcolors (>=1.11)"]
+format-nongpl = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3986-validator (>0.1.0)", "uri-template", "webcolors (>=1.11)"]
 
 [[package]]
 name = "jupyter-client"
 version = "7.3.1"
 description = "Jupyter protocol implementation and client libraries"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -600,7 +600,7 @@ test = ["codecov", "coverage", "ipykernel (>=6.5)", "ipython", "mypy", "pre-comm
 name = "jupyter-core"
 version = "4.10.0"
 description = "Jupyter core package. A base package on which Jupyter projects rely."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -615,7 +615,7 @@ test = ["ipykernel", "pre-commit", "pytest", "pytest-cov", "pytest-timeout"]
 name = "jupyterlab-pygments"
 version = "0.2.2"
 description = "Pygments theme using JupyterLab CSS variables"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -648,7 +648,7 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "llvmlite"
-version = "0.38.0"
+version = "0.38.1"
 description = "lightweight wrapper around basic LLVM functionality"
 category = "main"
 optional = false
@@ -717,15 +717,15 @@ python-versions = ">=3.6"
 name = "mistune"
 version = "0.8.4"
 description = "The fastest markdown parser in pure Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
 [[package]]
 name = "nbclient"
-version = "0.6.3"
+version = "0.6.4"
 description = "A client library for executing notebooks. Formerly nbconvert's ExecutePreprocessor."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7.0"
 
@@ -733,7 +733,7 @@ python-versions = ">=3.7.0"
 jupyter-client = ">=6.1.5"
 nbformat = ">=5.0"
 nest-asyncio = "*"
-traitlets = ">=5.0.0"
+traitlets = ">=5.2.2"
 
 [package.extras]
 sphinx = ["autodoc-traits", "mock", "moto", "myst-parser", "Sphinx (>=1.7)", "sphinx-book-theme"]
@@ -743,7 +743,7 @@ test = ["black", "check-manifest", "flake8", "ipykernel", "ipython (<8.0.0)", "i
 name = "nbconvert"
 version = "6.5.0"
 description = "Converting Jupyter Notebooks"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -776,7 +776,7 @@ webpdf = ["pyppeteer (>=1,<1.1)"]
 name = "nbformat"
 version = "5.4.0"
 description = "The Jupyter Notebook format"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -793,7 +793,7 @@ test = ["check-manifest", "testpath", "pytest", "pre-commit"]
 name = "nbsphinx"
 version = "0.8.8"
 description = "Jupyter Notebook Tools for Sphinx"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -809,7 +809,7 @@ traitlets = "*"
 name = "nest-asyncio"
 version = "1.5.5"
 description = "Patch asyncio to allow nested event loops"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.5"
 
@@ -823,7 +823,7 @@ python-versions = "*"
 
 [[package]]
 name = "numba"
-version = "0.55.1"
+version = "0.55.2"
 description = "compiling Python code using LLVM"
 category = "main"
 optional = false
@@ -831,15 +831,15 @@ python-versions = ">=3.7,<3.11"
 
 [package.dependencies]
 llvmlite = ">=0.38.0rc1,<0.39"
-numpy = ">=1.18,<1.22"
+numpy = ">=1.18,<1.23"
 
 [[package]]
 name = "numpy"
-version = "1.21.6"
+version = "1.22.4"
 description = "NumPy is the fundamental package for array computing with Python."
 category = "main"
 optional = false
-python-versions = ">=3.7,<3.11"
+python-versions = ">=3.8"
 
 [[package]]
 name = "packaging"
@@ -877,7 +877,7 @@ test = ["hypothesis (>=5.5.3)", "pytest (>=6.0)", "pytest-xdist (>=1.31)"]
 name = "pandocfilters"
 version = "1.5.0"
 description = "Utilities for writing pandoc filters in python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
@@ -943,7 +943,7 @@ python-versions = "*"
 
 [[package]]
 name = "pillow"
-version = "9.1.0"
+version = "9.1.1"
 description = "Python Imaging Library (Fork)"
 category = "main"
 optional = true
@@ -1006,14 +1006,14 @@ wcwidth = "*"
 
 [[package]]
 name = "psutil"
-version = "5.9.0"
+version = "5.9.1"
 description = "Cross-platform lib for process and system monitoring in Python."
 category = "dev"
 optional = false
-python-versions = ">=2.6, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
 [package.extras]
-test = ["ipaddress", "mock", "unittest2", "enum34", "pywin32", "wmi"]
+test = ["ipaddress", "mock", "enum34", "pywin32", "wmi"]
 
 [[package]]
 name = "ptyprocess"
@@ -1038,7 +1038,7 @@ tests = ["pytest"]
 name = "py"
 version = "1.11.0"
 description = "library with cross-python path, ini-parsing, io, code, log facilities"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
@@ -1060,7 +1060,7 @@ test = ["pytest"]
 
 [[package]]
 name = "pybtex-docutils"
-version = "1.0.1"
+version = "1.0.2"
 description = "A docutils backend for pybtex."
 category = "main"
 optional = false
@@ -1074,7 +1074,7 @@ pybtex = ">=0.16"
 name = "pycparser"
 version = "2.21"
 description = "C parser in Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
@@ -1088,28 +1088,30 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "pylint"
-version = "2.13.8"
+version = "2.14.0"
 description = "python code static checker"
 category = "dev"
 optional = false
-python-versions = ">=3.6.2"
+python-versions = ">=3.7.2"
 
 [package.dependencies]
-astroid = ">=2.11.3,<=2.12.0-dev0"
+astroid = ">=2.11.5,<=2.12.0-dev0"
 colorama = {version = "*", markers = "sys_platform == \"win32\""}
 dill = ">=0.2"
 isort = ">=4.2.5,<6"
 mccabe = ">=0.6,<0.8"
 platformdirs = ">=2.2.0"
 tomli = {version = ">=1.1.0", markers = "python_version < \"3.11\""}
+tomlkit = ">=0.10.1"
 typing-extensions = {version = ">=3.10.0", markers = "python_version < \"3.10\""}
 
 [package.extras]
-testutil = ["gitpython (>3)"]
+spelling = ["pyenchant (>=3.2,<4.0)"]
+testutils = ["gitpython (>3)"]
 
 [[package]]
 name = "pyparsing"
-version = "3.0.8"
+version = "3.0.9"
 description = "pyparsing module - Classes and methods to define and execute parsing grammars"
 category = "main"
 optional = false
@@ -1138,7 +1140,7 @@ python-versions = "*"
 name = "pyrsistent"
 version = "0.18.1"
 description = "Persistent/Functional/Immutable data structures"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -1220,7 +1222,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 name = "pywin32"
 version = "304"
 description = "Python for Window Extensions"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -1234,9 +1236,9 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "pyzmq"
-version = "22.3.0"
+version = "23.1.0"
 description = "Python bindings for 0MQ"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1280,7 +1282,7 @@ jupyter = ["ipywidgets (>=7.5.1,<8.0.0)"]
 
 [[package]]
 name = "scipy"
-version = "1.8.0"
+version = "1.8.1"
 description = "SciPy: Scientific Library for Python"
 category = "main"
 optional = false
@@ -1325,7 +1327,7 @@ python-versions = "*"
 name = "soupsieve"
 version = "2.3.2.post1"
 description = "A modern CSS selector implementation for Beautiful Soup."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1463,7 +1465,7 @@ test = ["pytest"]
 
 [[package]]
 name = "sqlalchemy"
-version = "1.4.36"
+version = "1.4.37"
 description = "Database Abstraction Library"
 category = "main"
 optional = true
@@ -1487,7 +1489,7 @@ mysql_connector = ["mysql-connector-python"]
 oracle = ["cx_oracle (>=7,<8)", "cx_oracle (>=7)"]
 postgresql = ["psycopg2 (>=2.7)"]
 postgresql_asyncpg = ["greenlet (!=0.4.17)", "asyncpg"]
-postgresql_pg8000 = ["pg8000 (>=1.16.6)"]
+postgresql_pg8000 = ["pg8000 (>=1.16.6,!=1.29.0)"]
 postgresql_psycopg2binary = ["psycopg2-binary"]
 postgresql_psycopg2cffi = ["psycopg2cffi"]
 pymysql = ["pymysql (<1)", "pymysql"]
@@ -1513,7 +1515,7 @@ tests = ["pytest", "typeguard", "pygments", "littleutils", "cython"]
 name = "tinycss2"
 version = "1.1.1"
 description = "A tiny CSS parser"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1540,24 +1542,32 @@ category = "main"
 optional = false
 python-versions = ">=3.7"
 
+[[package]]
+name = "tomlkit"
+version = "0.11.0"
+description = "Style preserving TOML library"
+category = "dev"
+optional = false
+python-versions = ">=3.6,<4.0"
+
 [[package]]
 name = "tornado"
 version = "6.1"
 description = "Tornado is a Python web framework and asynchronous networking library, originally developed at FriendFeed."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">= 3.5"
 
 [[package]]
 name = "traitlets"
-version = "5.1.1"
-description = "Traitlets Python configuration system"
+version = "5.2.2.post1"
+description = ""
 category = "main"
 optional = false
 python-versions = ">=3.7"
 
 [package.extras]
-test = ["pytest"]
+test = ["pre-commit", "pytest"]
 
 [[package]]
 name = "typing-extensions"
@@ -1610,7 +1620,7 @@ python-versions = "*"
 name = "webencodings"
 version = "0.5.1"
 description = "Character encoding aliases for legacy web content"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -1644,13 +1654,13 @@ testing = ["pytest (>=6)", "pytest-checkdocs (>=2.4)", "pytest-flake8", "pytest-
 
 [extras]
 box = ["banana-hep", "sqlalchemy", "pandas", "matplotlib"]
-docs = ["Sphinx", "sphinx-rtd-theme", "sphinxcontrib-bibtex"]
+docs = ["Sphinx", "sphinx-rtd-theme", "sphinxcontrib-bibtex", "nbsphinx"]
 mark = ["banana-hep", "sqlalchemy", "pandas", "matplotlib"]
 
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 [metadata.files]
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@@ -2814,6 +2835,10 @@ tomli = [
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     {file = "tomli-2.0.1.tar.gz", hash = "sha256:de526c12914f0c550d15924c62d72abc48d6fe7364aa87328337a31007fe8a4f"},
 ]
+tomlkit = [
+    {file = "tomlkit-0.11.0-py3-none-any.whl", hash = "sha256:0f4050db66fd445b885778900ce4dd9aea8c90c4721141fde0d6ade893820ef1"},
+    {file = "tomlkit-0.11.0.tar.gz", hash = "sha256:71ceb10c0eefd8b8f11fe34e8a51ad07812cb1dc3de23247425fbc9ddc47b9dd"},
+]
 tornado = [
     {file = "tornado-6.1-cp35-cp35m-macosx_10_9_x86_64.whl", hash = "sha256:d371e811d6b156d82aa5f9a4e08b58debf97c302a35714f6f45e35139c332e32"},
     {file = "tornado-6.1-cp35-cp35m-manylinux1_i686.whl", hash = "sha256:0d321a39c36e5f2c4ff12b4ed58d41390460f798422c4504e09eb5678e09998c"},
@@ -2858,8 +2883,8 @@ tornado = [
     {file = "tornado-6.1.tar.gz", hash = "sha256:33c6e81d7bd55b468d2e793517c909b139960b6c790a60b7991b9b6b76fb9791"},
 ]
 traitlets = [
-    {file = "traitlets-5.1.1-py3-none-any.whl", hash = "sha256:2d313cc50a42cd6c277e7d7dc8d4d7fedd06a2c215f78766ae7b1a66277e0033"},
-    {file = "traitlets-5.1.1.tar.gz", hash = "sha256:059f456c5a7c1c82b98c2e8c799f39c9b8128f6d0d46941ee118daace9eb70c7"},
+    {file = "traitlets-5.2.2.post1-py3-none-any.whl", hash = "sha256:1530d04badddc6a73d50b7ee34667d4b96914da352109117b4280cb56523a51b"},
+    {file = "traitlets-5.2.2.post1.tar.gz", hash = "sha256:74803a1baa59af70f023671d86d5c7a834c931186df26d50d362ee6a1ff021fd"},
 ]
 typing-extensions = [
     {file = "typing_extensions-4.2.0-py3-none-any.whl", hash = "sha256:6657594ee297170d19f67d55c05852a874e7eb634f4f753dbd667855e07c1708"},
diff --git a/src/eko/anomalous_dimensions/as4/__init__.py b/src/eko/anomalous_dimensions/as4/__init__.py
new file mode 100644
index 000000000..343bf7a66
--- /dev/null
+++ b/src/eko/anomalous_dimensions/as4/__init__.py
@@ -0,0 +1,9 @@
+# -*- coding: utf-8 -*-
+"""This subpackage contains the |N3LO| Altarelli-Parisi splitting kernels.
+
+For further documentation see :doc:`N3LO anomalous dimensions <../../../theory/N3LO_ad>`
+
+"""
+from .gnsm import gamma_nsm
+from .gnsp import gamma_nsp
+from .gnsv import gamma_nsv
diff --git a/src/eko/anomalous_dimensions/as4/gnsm.py b/src/eko/anomalous_dimensions/as4/gnsm.py
new file mode 100644
index 000000000..e4bdb719b
--- /dev/null
+++ b/src/eko/anomalous_dimensions/as4/gnsm.py
@@ -0,0 +1,212 @@
+# -*- coding: utf-8 -*-
+"""This module contains the anomalous dimension :math:`\\gamma_{ns,-}^{(3)}`
+
+"""
+import numba as nb
+
+from ...constants import CF
+from ...harmonics.constants import zeta3
+from ...harmonics.log_functions import lm11m1, lm12m1, lm13m1
+
+
+@nb.njit(cache=True)
+def gamma_ns_nf3(n, sx):
+    """Implements the common part proportional to :math:`nf^3`,
+    of :math:`\\gamma_{ns,+}^{(3)},\\gamma_{ns,-}^{(3)},\\gamma_{ns,v}^{(3)}`.
+    The expression is copied exact from Eq. 3.6. of :cite:`Davies:2016jie`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_ns_nf3 : complex
+        |N3LO| non-singlet anomalous dimension :math:`\\gamma_{ns}^{(3)}|_{nf^3}`
+
+    """
+    S1 = sx[0][0]
+    S2 = sx[1][0]
+    S3 = sx[2][0]
+    S4 = sx[3][0]
+    eta = 1 / n * 1 / (n + 1)
+    g_ns_nf3 = CF * (
+        -32 / 27 * zeta3 * eta
+        - 16 / 9 * zeta3
+        - 16 / 27 * eta**4
+        - 16 / 81 * eta**3
+        + 80 / 27 * eta**2
+        - 320 / 81 * eta
+        + 32 / 27 * 1 / (n + 1) ** 4
+        + 128 / 27 * 1 / (n + 1) ** 2
+        + 64 / 27 * S1 * zeta3
+        - 32 / 81 * S1
+        - 32 / 81 * S2
+        - 160 / 81 * S3
+        + 32 / 27 * S4
+        + 131 / 81
+    )
+    return g_ns_nf3
+
+
+@nb.njit(cache=True)
+def gamma_nsm_nf2(n, sx):
+    """Implements the parametrized valence-like non-singlet part proportional to :math:`nf^2`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_ns_nf2 : complex
+        |N3LO| valence-like non-singlet anomalous dimension
+        :math:`\\gamma_{ns,-}^{(3)}|_{nf^2}`
+
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        -193.85692903712987
+        - 18.962962962962962 / n**5
+        + 99.1604938271605 / n**4
+        - 226.44075306899038 / n**3
+        + 395.60497732877303 / n**2
+        + 278.2205375565073 / n
+        + 433.6813853974481 / (1.0 + n) ** 3
+        + 77.59871714838847 / (1.0 + n) ** 2
+        - 96.10213330829208 / (3.0 + n)
+        + 195.5772257829161 * S1
+        - (680.3854257809448 * S1) / n**2
+        + (26.68861454046639 * S1) / n
+        - 76.56139620361596 * Lm11m1
+        + 126.58793206934183 * Lm12m1
+        + 13.001858308748329 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsm_nf1(n, sx):
+    """Implements the parametrized valence-like non-singlet part proportional to :math:`nf^1`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_ns_nf1 : complex
+        |N3LO| valence-like non-singlet anomalous dimension
+        :math:`\\gamma_{ns,-}^{(3)}|_{nf^1}`
+
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        5549.7951398549685
+        - 126.41975308641975 / n**6
+        + 752.1975308641976 / n**5
+        - 2253.1105700880144 / n**4
+        + 5247.1769880520205 / n**3
+        - 8769.153217295072 / n**2
+        - 5834.355552528428 / n
+        - 9279.55898142154 / (1.0 + n) ** 3
+        - 2551.4893186377476 / (1.0 + n) ** 2
+        + 2632.8150611003307 / (3.0 + n)
+        - 5171.916129085788 * S1
+        + (16105.088692447685 * S1) / n**2
+        - (2741.830025124657 * S1) / n
+        + 2990.9289986960293 * Lm11m1
+        - 2357.8218160665683 * Lm12m1
+        - 151.90951695259517 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsm_nf0(n, sx):
+    """Implements the parametrized valence-like non-singlet part proportional to :math:`nf^0`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+        g_ns_nf0 : complex
+            |N3LO| valence-like non-singlet anomalous dimension
+            :math:`\\gamma_{ns,-}^{(3)}|_{nf^0}`
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        -23383.08164724965
+        - 252.8395061728395 / n**7
+        + 1580.2469135802469 / n**6
+        - 5806.800104704373 / n**5
+        + 14899.91711929902 / n**4
+        - 28546.38768506619 / n**3
+        + 50759.65541232588 / n**2
+        + 21477.757730073346 / n
+        - 14396.446623479116 / (1.0 + n) ** 3
+        - 83049.79603015777 / (1.0 + n) ** 2
+        - 13242.729433996457 / (3.0 + n)
+        + 20702.353028966703 * S1
+        - (39225.8841951996 * S1) / n**2
+        + (16950.937339235086 * S1) / n
+        - 32830.98964964463 * Lm11m1
+        - 13907.903486384817 * Lm12m1
+        - 4337.4357337052215 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsm(n, nf, sx):
+    """Computes the |N3LO| valence-like non-singlet anomalous dimension.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    nf : int
+        Number of active flavors
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    gamma_nsp : complex
+        |N3LO| valence-like non-singlet anomalous dimension
+        :math:`\\gamma_{ns,-}^{(3)}(N)`
+
+    See Also
+    --------
+    gamma_nsm_nf0: :math:`\\gamma_{ns,-}^{(3)}|_{nf^0}`
+    gamma_nsm_nf1: :math:`\\gamma_{ns,-}^{(3)}|_{nf^1}`
+    gamma_nsm_nf2: :math:`\\gamma_{ns,-}^{(3)}|_{nf^2}`
+    gamma_ns_nf3: :math:`\\gamma_{ns}^{(3)}|_{nf^3}`
+
+    """
+    return (
+        gamma_nsm_nf0(n, sx)
+        + nf * gamma_nsm_nf1(n, sx)
+        + nf**2 * gamma_nsm_nf2(n, sx)
+        + nf**3 * gamma_ns_nf3(n, sx)
+    )
diff --git a/src/eko/anomalous_dimensions/as4/gnsp.py b/src/eko/anomalous_dimensions/as4/gnsp.py
new file mode 100644
index 000000000..31f20f62d
--- /dev/null
+++ b/src/eko/anomalous_dimensions/as4/gnsp.py
@@ -0,0 +1,166 @@
+# -*- coding: utf-8 -*-
+"""This module contains the anomalous dimension :math:`\\gamma_{ns,+}^{(3)}`
+
+"""
+import numba as nb
+
+from ...harmonics.log_functions import lm11m1, lm12m1, lm13m1
+from .gnsm import gamma_ns_nf3
+
+
+@nb.njit(cache=True)
+def gamma_nsp_nf2(n, sx):
+    """Implements the parametrized singlet-like non-singlet part proportional to :math:`nf^2`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_nsp_nf2 : complex
+        |N3LO| singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+}^{(3)}|_{nf^2}`
+
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        -193.85479604848626
+        - 18.962962962962962 / n**5
+        + 99.1604938271605 / n**4
+        - 226.44075306899038 / n**3
+        + 395.60497732877303 / n**2
+        + 278.2205375565073 / n
+        + 351.0191873322347 / (1.0 + n) ** 3
+        - 115.83082177850747 / (1.0 + n) ** 2
+        - 97.28108136993096 / (2.0 + n)
+        + 195.5772257829161 * S1
+        - (617.3467445575227 * S1) / n**2
+        + (26.68861454046639 * S1) / n
+        - 179.1958949029147 * Lm11m1
+        + 78.89342505600462 * Lm12m1
+        + 7.093169138453943 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsp_nf1(n, sx):
+    """Implements the parametrized singlet-like non-singlet part proportional to :math:`nf^1`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_nsp_nf1 : complex
+        |N3LO| singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+}^{(3)}|_{nf^1}`
+
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        5550.063827367692
+        - 126.41975308641975 / n**6
+        + 752.1975308641976 / n**5
+        - 2253.1105700880144 / n**4
+        + 5247.1769880520205 / n**3
+        - 8769.153217295072 / n**2
+        - 5834.355552528428 / n
+        - 7856.698193404992 / (1.0 + n) ** 3
+        - 1780.0619991736955 / (1.0 + n) ** 2
+        + 2565.962271083018 / (2.0 + n)
+        - 5171.916129085788 * S1
+        + (15756.961591759926 * S1) / n**2
+        - (2741.830025124657 * S1) / n
+        + 4352.896032770554 * Lm11m1
+        - 2110.3938915736526 * Lm12m1
+        - 204.23062235898342 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsp_nf0(n, sx):
+    """Implements the parametrized singlet-like non-singlet part proportional to :math:`nf^0`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_nsp_nf0 : complex
+        |N3LO| singlet-like non-singlet anomalous dimension :math:`\\gamma_{ns,+}^{(3)}|_{nf^0}`
+
+    """
+    S1 = sx[0][0]
+    Lm11m1 = lm11m1(n, S1)
+    Lm12m1 = lm12m1(n, S1, sx[1][0])
+    Lm13m1 = lm13m1(n, S1, sx[1][0], sx[2][0])
+    return (
+        -23391.854890259732
+        - 252.8395061728395 / n**7
+        + 1580.2469135802469 / n**6
+        - 5806.800104704373 / n**5
+        + 14899.91711929902 / n**4
+        - 28546.38768506619 / n**3
+        + 50759.65541232588 / n**2
+        + 21477.757730073346 / n
+        + 26932.571453853965 / (1.0 + n) ** 3
+        - 17764.722568307116 / (1.0 + n) ** 2
+        - 10913.245002601085 / (2.0 + n)
+        + 20702.353028966703 * S1
+        - (66520.50722889033 * S1) / n**2
+        + (16950.937339235086 * S1) / n
+        - 31046.58355660925 * Lm11m1
+        - 90.76127492188596 * Lm12m1
+        - 497.92637248421335 * Lm13m1
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsp(n, nf, sx):
+    """Computes the |N3LO| singlet-like non-singlet anomalous dimension.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    nf : int
+        Number of active flavors
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    gamma_nsp : complex
+        |N3LO| singlet-like non-singlet anomalous dimension
+        :math:`\\gamma_{ns,+}^{(3)}(N)`
+
+    See Also
+    --------
+    gamma_nsp_nf0: :math:`\\gamma_{ns,+}^{(3)}|_{nf^0}`
+    gamma_nsp_nf1: :math:`\\gamma_{ns,+}^{(3)}|_{nf^1}`
+    gamma_nsp_nf2: :math:`\\gamma_{ns,+}^{(3)}|_{nf^2}`
+    gamma_ns_nf3: :math:`\\gamma_{ns}^{(3)}|_{nf^3}`
+
+    """
+    return (
+        gamma_nsp_nf0(n, sx)
+        + nf * gamma_nsp_nf1(n, sx)
+        + nf**2 * gamma_nsp_nf2(n, sx)
+        + nf**3 * gamma_ns_nf3(n, sx)
+    )
diff --git a/src/eko/anomalous_dimensions/as4/gnsv.py b/src/eko/anomalous_dimensions/as4/gnsv.py
new file mode 100644
index 000000000..c401d3da9
--- /dev/null
+++ b/src/eko/anomalous_dimensions/as4/gnsv.py
@@ -0,0 +1,251 @@
+# -*- coding: utf-8 -*-
+"""This module contains the anomalous dimension :math:`\\gamma_{ns,v}^{(3)}`
+
+"""
+import numba as nb
+
+from .gnsm import gamma_nsm
+
+
+@nb.njit(cache=True)
+def gamma_nss_nf2(n, sx):
+    """Implements the sea non-singlet part proportional to :math:`nf^2`
+    as in Eq. 3.5 of :cite:`Davies:2016jie`.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_nss_nf2 : complex
+        |N3LO| sea non-singlet anomalous dimension :math:`\\gamma_{ns,s}^{(3)}|_{nf^2}`
+
+    """
+    S1, _ = sx[0]
+    S2, Sm2 = sx[1]
+    S3, _, _, Sm21, _, Sm3 = sx[2]
+    S4, S31, _, Sm22, Sm211, Sm31, Sm4 = sx[3]
+    return (
+        160
+        / 27
+        * (
+            40 / (1 + n) ** 6
+            - 8 / (n**6 * (1 + n) ** 6)
+            - 16 / (1 + n) ** 5
+            - 24 / (n**5 * (1 + n) ** 5)
+            + 108 / (1 + n) ** 4
+            + 36 / (n**4 * (1 + n) ** 4)
+            + 112 / (1 + n) ** 3
+            + 168 / (n**3 * (1 + n) ** 3)
+            + 144 / (1 + n) ** 2
+            + 174 / (n**2 * (1 + n) ** 2)
+            - 64 / (n * (1 + n))
+            - 32 / (3 * (2 + n) ** 2)
+            - 208 / (3 * (-1 + n) * (2 + n))
+            + (
+                48 / (1 + n) ** 5
+                + 4 / (n**5 * (1 + n) ** 5)
+                + 6 / (1 + n) ** 4
+                - 24 / (n**4 * (1 + n) ** 4)
+                + 76 / (1 + n) ** 3
+                - 162 / (n**3 * (1 + n) ** 3)
+                + 50 / (1 + n) ** 2
+                - 164 / (n**2 * (1 + n) ** 2)
+                - 285 / (2 * n * (1 + n))
+                + 32 / (3 * (2 + n) ** 2)
+                + 304 / (3 * (-1 + n) * (2 + n))
+            )
+            * S1
+            - (
+                -(6 / (n**4 * (1 + n) ** 4))
+                - 23 / (n**3 * (1 + n) ** 3)
+                - 28 / (n**2 * (1 + n) ** 2)
+                - 13 / (n * (1 + n))
+                + 16 / ((-1 + n) * (2 + n))
+            )
+            * (-S2 + 2 * (S1**2 + S2))
+            + (
+                -(8 / (1 + n) ** 3)
+                - 6 / (n**3 * (1 + n) ** 3)
+                - 6 / (1 + n) ** 2
+                - 7 / (n**2 * (1 + n) ** 2)
+                - 9 / (n * (1 + n))
+                + 16 / ((-1 + n) * (2 + n))
+            )
+            * S3
+            + (
+                -(12 / (n**2 * (1 + n) ** 2))
+                - 22 / (n * (1 + n))
+                + 32 / ((-1 + n) * (2 + n))
+            )
+            * S4
+            + (
+                24 / (1 + n) ** 4
+                - 8 / (1 + n) ** 3
+                + 28 / (n**3 * (1 + n) ** 3)
+                + 28 / (1 + n) ** 2
+                + 104 / (n**2 * (1 + n) ** 2)
+                + 91 / (n * (1 + n))
+                - 32 / (3 * (2 + n) ** 2)
+                - 304 / (3 * (-1 + n) * (2 + n))
+            )
+            * Sm2
+            + (
+                -(8 / (n**3 * (1 + n) ** 3))
+                - 16 / (1 + n) ** 2
+                - 28 / (n**2 * (1 + n) ** 2)
+                - 76 / (n * (1 + n))
+                + 32 / (3 * (2 + n) ** 2)
+                + 352 / (3 * (-1 + n) * (2 + n))
+            )
+            * Sm21
+            - (-(8 / (n * (1 + n))) + 16 / ((-1 + n) * (2 + n)))
+            * (S4 + Sm2**2 + 4 * Sm211 - Sm22)
+            + (
+                -(16 / (1 + n) ** 3)
+                - 4 / (n**3 * (1 + n) ** 3)
+                + 4 / (1 + n) ** 2
+                + 14 / (n**2 * (1 + n) ** 2)
+                + 58 / (n * (1 + n))
+                - 32 / (3 * (2 + n) ** 2)
+                - 256 / (3 * (-1 + n) * (2 + n))
+            )
+            * Sm3
+            + (
+                32 / (1 + n) ** 3
+                + 8 / (1 + n) ** 2
+                - 56 / (n**2 * (1 + n) ** 2)
+                - 112 / (n * (1 + n))
+                + 32 / (3 * (2 + n) ** 2)
+                + 352 / (3 * (-1 + n) * (2 + n))
+            )
+            * (S1 * Sm2 - Sm21 + Sm3)
+            + (
+                -(4 / (n**2 * (1 + n) ** 2))
+                - 10 / (n * (1 + n))
+                + 16 / ((-1 + n) * (2 + n))
+            )
+            * (
+                -S1 * S3
+                + S31
+                - S4
+                + 2 * Sm31
+                - 4 * (S1 * Sm21 - 2 * Sm211 + Sm22 + Sm31)
+                + Sm4
+                + 2 * (S1 * Sm3 - Sm31 + Sm4)
+            )
+            - (
+                -(8 / (n**2 * (1 + n) ** 2))
+                - 12 / (n * (1 + n))
+                + 16 / ((-1 + n) * (2 + n))
+            )
+            * (
+                S31
+                - S2 * Sm2
+                + Sm22
+                - Sm4
+                + 4
+                * (
+                    S2 * Sm2
+                    - 1 / 2 * (S1**2 + S2) * Sm2
+                    + Sm211
+                    - Sm22
+                    + S1 * (S1 * Sm2 - Sm21 + Sm3)
+                    - Sm31
+                    + Sm4
+                )
+            )
+        )
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nss_nf1(n, sx):
+    """Implements the sea non-singlet part proportional to :math:`nf^1`.
+    The expression is the average of the Mellin transform
+    of Eq. 4.19, 4.20 of :cite:`Moch:2017uml`
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+    g_nss_nf1 : complex
+        |N3LO| sea non-singlet anomalous dimension :math:`\\gamma_{ns,s}^{(3)}|_{nf^1}`
+
+    """
+    S1 = sx[0][0]
+    S2 = sx[1][0]
+    S3 = sx[2][0]
+    return (
+        3803.04 / n**6
+        - 3560.16 / n**5
+        + 1042.89 / n**4
+        - 3569.65 / n**3
+        + 3337.715 / n**2
+        - 1266.77 / n
+        - 13.18641 / (1 + n) ** 4
+        + 60.4 / (1 + n) ** 3
+        - 1743.225 / (1 + n)
+        + 127.315 / (1.0 + n) ** 2
+        + 4447.245 / (2 + n)
+        - 1437.25 / (3 + n)
+        - (127.315 * S1) / n
+        - (13.18641 * S1) / (1 + n) ** 3
+        + (60.4 * S1) / (1 + n) ** 2
+        + (127.315 * S1) / (1.0 + n) ** 2
+        + (127.315 * n * S1) / (1.0 + n) ** 2
+        - (30.2 * S1**2) / n
+        - (6.5932 * S1**2) / (1 + n) ** 2
+        + (30.2 * S1**2) / (1 + n)
+        + (2.197735 * S1**3) / n
+        - (2.197735 * S1**3) / (1 + n)
+        - (30.2 * S2) / n
+        - (6.593205 * S2) / (1 + n) ** 2
+        + (30.2 * S2) / (1 + n)
+        + (6.593205 * S1 * S2) / n
+        - (6.593205 * S1 * S2) / (1 + n)
+        + (4.39547 * S3) / n
+        - (4.39547 * S3) / (1 + n)
+    )
+
+
+@nb.njit(cache=True)
+def gamma_nsv(n, nf, sx):
+    """Computes the |N3LO| valence non-singlet anomalous dimension.
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    nf : int
+        Number of active flavors
+    sx : list
+        harmonic sums cache
+
+    Returns
+    -------
+        gamma_nsv : complex
+            |N3LO| valence non-singlet anomalous dimension
+            :math:`\\gamma_{ns,v}^{(3)}(N)`
+
+    See Also
+    --------
+    gamma_nsm: :math:`\\gamma_{ns,-}^{(3)}`
+    gamma_nss_nf1: :math:`\\gamma_{ns,s}^{(3)}|_{nf^1}`
+    gamma_nss_nf2: :math:`\\gamma_{ns,s}^{(3)}|_{nf^2}`
+
+    """
+    return (
+        gamma_nsm(n, nf, sx)
+        + nf * gamma_nss_nf1(n, sx)
+        + nf**2 * gamma_nss_nf2(n, sx)
+    )
diff --git a/src/eko/harmonics/__init__.py b/src/eko/harmonics/__init__.py
index 087ea60f0..b50860077 100644
--- a/src/eko/harmonics/__init__.py
+++ b/src/eko/harmonics/__init__.py
@@ -1,7 +1,7 @@
 # -*- coding: utf-8 -*-
 """
 This module contains some harmonics sum.
-Defintion are coming from :cite:`MuselliPhD,Bl_mlein_2000,Blumlein:2009ta`
+Definitions are coming from :cite:`MuselliPhD,Bl_mlein_2000,Blumlein:2009ta`
 """
 import numba as nb
 import numpy as np
diff --git a/src/eko/harmonics/g_functions.py b/src/eko/harmonics/g_functions.py
index 220dc95ed..63e4392f0 100644
--- a/src/eko/harmonics/g_functions.py
+++ b/src/eko/harmonics/g_functions.py
@@ -1,6 +1,6 @@
 # -*- coding: utf-8 -*-
 """
-Implemtations of some mellin transformations :cite:`MuselliPhD`
+Implementations of some mellin transformations :cite:`MuselliPhD`
 appearing in the analytic continuation of harmonics sums of weight = 3,4.
 """
 import numba as nb
diff --git a/src/eko/harmonics/log_functions.py b/src/eko/harmonics/log_functions.py
new file mode 100644
index 000000000..f62dde3ea
--- /dev/null
+++ b/src/eko/harmonics/log_functions.py
@@ -0,0 +1,90 @@
+# -*- coding: utf-8 -*-
+"""Implementation of some Mellin transformation of functions as:
+
+    :math:`(1-x)\\ln^k(1-x), \\quad k = 1,2,3`
+
+"""
+import numba as nb
+
+from .constants import zeta3
+
+
+@nb.njit(cache=True)
+def lm11m1(n, S1):
+    """Mellin transform of :math:`(1-x)\\ln(1-x)`
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    S1:  complex
+        Harmonic sum :math:`S_{1}(N)`
+
+    Returns
+    -------
+    complex
+
+    """
+    return 1 / (1 + n) ** 2 - S1 / (1 + n) ** 2 - S1 / (n * (1 + n) ** 2)
+
+
+@nb.njit(cache=True)
+def lm12m1(n, S1, S2):
+    """Mellin transform of :math:`(1-x)\\ln^2(1-x)`
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    S1:  complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2:  complex
+        Harmonic sum :math:`S_{2}(N)`
+
+    Returns
+    -------
+    complex
+
+    """
+    return (
+        -2 / (1 + n) ** 3
+        - (2 * S1) / (1 + n) ** 2
+        + S1**2 / n
+        - S1**2 / (1 + n)
+        + S2 / n
+        - S2 / (1 + n)
+    )
+
+
+@nb.njit(cache=True)
+def lm13m1(n, S1, S2, S3):
+    """Mellin transform of :math:`(1-x)\\ln^3(1-x)`
+
+    Parameters
+    ----------
+    n : complex
+        Mellin moment
+    S1:  complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2:  complex
+        Harmonic sum :math:`S_{2}(N)`
+    S3:  complex
+        Harmonic sum :math:`S_{3}(N)`
+
+    Returns
+    -------
+    complex
+
+    """
+    return (
+        6 / (1 + n) ** 4
+        + (6 * S1) / (1 + n) ** 3
+        + (3 * S1**2) / (1 + n) ** 2
+        - S1**3 / n
+        + S1**3 / (1 + n)
+        + (3 * S2) / (1 + n) ** 2
+        - (3 * S1 * S2) / n
+        + (3 * S1 * S2) / (1 + n)
+        - (2 * (6 * (2 * S3 - 2 * zeta3) + zeta3)) / n
+        + (2 * (6 * (2 * S3 - 2 * zeta3) + zeta3)) / (1 + n)
+    )
diff --git a/src/eko/matching_conditions/operator_matrix_element.py b/src/eko/matching_conditions/operator_matrix_element.py
index bbf62c15c..8768d23c6 100644
--- a/src/eko/matching_conditions/operator_matrix_element.py
+++ b/src/eko/matching_conditions/operator_matrix_element.py
@@ -48,22 +48,19 @@ def compute_harmonics_cache(n, order, is_singlet):
 
     """
     # max harmonics sum weight for each qcd order
-    max_weight = {1: 2, 2: 3, 3: 3}
-    # _N3LO_ max_weight = {1: 2, 2: 3, 3: 5}
+    max_weight = {1: 2, 2: 3, 3: 5}
     # max number of harmonics sum of a given weight for each qcd order
-    n_max_sums_weight = {1: 1, 2: 3, 3: 3}
-    # _N3LO_ n_max_sums_weight = {1: 1, 2: 3, 3: 7}
+    n_max_sums_weight = {1: 1, 2: 3, 3: 7}
     sx = harmonics.base_harmonics_cache(
         n, is_singlet, max_weight[order], n_max_sums_weight[order]
     )
-    # _N3LO_ if order == 2:
-    if order >= 2:
+    if order == 2:
         # Add Sm21 to cache
         sx[2, 1] = harmonics.Sm21(n, sx[0, 0], sx[0, -1], is_singlet)
-    # _N3LO_ if order == 3:
-    #     # Add weight 3 and 4 to cache
-    #     sx[2, 1:-2] = harmonics.s3x(n, sx[:, 0], sx[:, -1], is_singlet)
-    #     sx[3, 1:-1] = harmonics.s4x(n, sx[:, 0], sx[:, -1], is_singlet)
+    if order == 3:
+        # Add weight 3 and 4 to cache
+        sx[2, 1:-2] = harmonics.s3x(n, sx[:, 0], sx[:, -1], is_singlet)
+        sx[3, 1:-1] = harmonics.s4x(n, sx[:, 0], sx[:, -1], is_singlet)
     # return list of list keeping the non zero values
     return [[el for el in sx_list if el != 0] for sx_list in sx]
 
diff --git a/tests/eko/test_ad_as3.py b/tests/eko/test_ad_as3.py
index 8348712fa..e0ce0d2cc 100644
--- a/tests/eko/test_ad_as3.py
+++ b/tests/eko/test_ad_as3.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# Test NNLO anomalous dims
+# Test NNLO anomalous dimensions
 import numpy as np
 
 import eko.anomalous_dimensions.as3 as ad_as3
diff --git a/tests/eko/test_ad_as4.py b/tests/eko/test_ad_as4.py
new file mode 100644
index 000000000..4b2c2d1d6
--- /dev/null
+++ b/tests/eko/test_ad_as4.py
@@ -0,0 +1,177 @@
+# -*- coding: utf-8 -*-
+# Test N3LO anomalous dimensions
+import numpy as np
+
+from eko.anomalous_dimensions.as4 import gnsm, gnsp, gnsv
+from eko.constants import CA, CF
+
+# TODO: move this method  out of matching conditions if it will be used
+# also here...
+from eko.matching_conditions.operator_matrix_element import compute_harmonics_cache
+
+NF = 5
+
+
+def test_quark_number_conservation():
+    N = 1
+    sx_cache = compute_harmonics_cache(N, 3, False)
+
+    # (ns,s)
+    # the exact expression (nf^2 part) has an nonphysical pole at N=1,
+    # see also :cite:`Moch:2017uml` and :cite:`Davies:2016jie` eq 3.5
+    # where the \nu term is present.
+    # This should cancel when doing the limit, since the given moment for N=1 is 0
+    # np.testing.assert_allclose(gamma_nsv(N, NF, sx_cache), 0, rtol=3e-7)
+
+    # nf^1 part
+    np.testing.assert_allclose(gnsv.gamma_nss_nf1(N, sx_cache), 0.000400625, atol=2e-6)
+
+    # (ns,-)
+    # nf^3 part
+    np.testing.assert_allclose(gnsp.gamma_ns_nf3(N, sx_cache), 0, atol=3e-15)
+    # nf^2 part
+    np.testing.assert_allclose(gnsm.gamma_nsm_nf2(N, sx_cache), 0, atol=3e-13)
+    # nf^1 part
+    np.testing.assert_allclose(gnsm.gamma_nsm_nf1(N, sx_cache), 0, atol=2e-11)
+    # nf^0 part
+    np.testing.assert_allclose(gnsm.gamma_nsm_nf0(N, sx_cache), 0, atol=2e-10)
+    # total
+    np.testing.assert_allclose(gnsm.gamma_nsm(N, NF, sx_cache), 0, atol=1e-10)
+
+
+def test_non_singlet_reference_moments():
+
+    NF = 4
+    nsm_nf4_refs = [
+        4322.890485339998,
+        5491.581109692005,
+        6221.256799360004,
+        6774.606221595994,
+        7229.056043916002,
+        7618.358743427995,
+        7960.658678124,
+    ]
+    nss_nf4_refs = [
+        50.10532524,
+        39.001939964,
+        21.141505811200002,
+        12.4834195012,
+        8.0006134908,
+        5.4610639744,
+        3.9114290952,
+        2.90857799,
+    ]
+    for N in [3.0, 5.0, 7.0, 9.0, 11.0, 13.0, 15.0, 17.0]:
+        sx_cache = compute_harmonics_cache(N, 3, False)
+        idx = int((N - 3) / 2)
+        if N != 17:
+            np.testing.assert_allclose(
+                gnsm.gamma_nsm(N, NF, sx_cache), nsm_nf4_refs[idx]
+            )
+            np.testing.assert_allclose(
+                gnsv.gamma_nsv(N, NF, sx_cache), nss_nf4_refs[idx] + nsm_nf4_refs[idx]
+            )
+        gamma_nss = (
+            gnsv.gamma_nss_nf1(N, sx_cache) * NF
+            + gnsv.gamma_nss_nf2(N, sx_cache) * NF**2
+        )
+        np.testing.assert_allclose(gamma_nss, nss_nf4_refs[idx], atol=4e-4)
+
+
+def test_singlet_reference_moments():
+    NF = 4
+    nsp_nf4_refs = [
+        3679.6690577439995,
+        5066.339235808004,
+        5908.005605364002,
+        6522.700744595994,
+        7016.383458928004,
+        7433.340927783997,
+        7796.397038483998,
+        8119.044600816003,
+    ]
+    for N in [2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0]:
+        sx_cache = compute_harmonics_cache(N, 3, True)
+        np.testing.assert_allclose(
+            gnsp.gamma_nsp(N, NF, sx_cache), nsp_nf4_refs[int((N - 2) / 2)]
+        )
+
+
+def test_diff_pm_nf2():
+    # Test deltaB3: diff g_{ns,-} - g_{ns,+} prop to nf^2
+    # Note that discrepancy for low moments is higher due to
+    # oscillating behavior which is not captured by our parametrization
+    def deltaB3(n, sx):
+        """
+        Implementation of Eq. 3.4 of :cite:`Davies:2016jie`.
+
+        Parameters
+        ----------
+            n : complex
+                Mellin moment
+            sx : list
+                harmonic sums cache
+
+        Returns
+        -------
+            B_3m : complex
+                |N3LO| valence-like non-singlet anomalous dimension part
+                proportional to :math:`C_F (C_A - 2 C_F) nf^2`
+        """
+        S1, _ = sx[0]
+        S2, Sm2 = sx[1]
+        deltaB = (
+            -(1 / (729 * n**5 * (1 + n) ** 5))
+            * CF
+            * (CA - 2 * CF)
+            * 9
+            * 16
+            * (
+                -54
+                - 60 * n
+                + 211 * n**2
+                + 367 * n**3
+                + 73 * n**4
+                - 255 * n**5
+                - 475 * n**6
+                - 138 * n**7
+                - 39 * n**8
+                - 36 * n**2 * (1 + n) ** 2 * (1 + 2 * n + 2 * n**2) * Sm2
+                + 6
+                * n
+                * (
+                    -12
+                    - 22 * n
+                    + 29 * n**2
+                    + 116 * n**3
+                    + 147 * n**4
+                    + 79 * n**5
+                    + 12 * n**6
+                    + 3 * n**7
+                )
+                * S1
+                - 36 * n**2 * (1 + n) ** 2 * (1 + 2 * n + 2 * n**2) * S1**2
+                - 36 * n**2 * S2
+                - 144 * n**3 * S2
+                - 252 * n**4 * S2
+                - 216 * n**5 * S2
+                - 72 * n**6 * S2
+            )
+        )
+        return deltaB
+
+    diff = []
+    ref_vals = []
+    for N in range(10, 51):
+        sx_cache = compute_harmonics_cache(N, 3, not bool(N % 2))
+        diff.append(gnsp.gamma_nsp_nf2(N, sx_cache) - gnsm.gamma_nsm_nf2(N, sx_cache))
+        ref_vals.append(deltaB3(N, sx_cache))
+    np.testing.assert_allclose(diff, ref_vals, atol=5e-4)
+
+    diff = []
+    ref_vals = []
+    for N in range(4, 10):
+        sx_cache = compute_harmonics_cache(N, 3, not bool(N % 2))
+        diff.append(gnsp.gamma_nsp_nf2(N, sx_cache) - gnsm.gamma_nsm_nf2(N, sx_cache))
+        ref_vals.append(deltaB3(N, sx_cache))
+    np.testing.assert_allclose(diff, ref_vals, atol=2e-2)