diff --git a/src/sage/graphs/generators/distance_regular.pyx b/src/sage/graphs/generators/distance_regular.pyx
index 570012ac3fc..4b5c8ac7657 100644
--- a/src/sage/graphs/generators/distance_regular.pyx
+++ b/src/sage/graphs/generators/distance_regular.pyx
@@ -488,34 +488,58 @@ def shortened_000_111_extended_binary_Golay_code_graph():
     EXAMPLES::
 
         sage: # long time, needs sage.modules sage.rings.finite_rings
-        sage: G = graphs.shortened_000_111_extended_binary_Golay_code_graph()   # 25 s
+        sage: G = graphs.shortened_000_111_extended_binary_Golay_code_graph()
         sage: G.is_distance_regular(True)
         ([21, 20, 16, 9, 2, 1, None], [None, 1, 2, 3, 16, 20, 21])
 
     ALGORITHM:
 
-    Compute the extended binary Golay code. Compute its subcode whose codewords
-    start with 000 or 111. Remove the first 3 entries from all the codewords
-    from the new linear code and compute its coset graph.
+    The vertices and edges of this graph have been precomputed and
+    pickled, so truthfully, we just unpickle them and pass them to the
+    Graph constructor. But the algorithm used to compute those
+    vertices and edges in the first place is,
 
-    REFERENCES:
-
-    Description and construction of this graph can be found in [BCN1989]_ p. 365.
-    """
-    from sage.coding.linear_code import LinearCode
-
-    code = codes.GolayCode(GF(2))
-    C_basis = code.basis()
+    #. Compute the extended binary Golay code.
+    #. Compute its subcode whose codewords start with 000 or 111.
+    #. Remove the first 3 entries from all the codewords from the
+       new linear code and compute its coset graph.
 
-    # now special shortening
-    v = C_basis[0] + C_basis[1] + C_basis[2]  # v has 111 at the start
-    C_basis = C_basis[3:]
-    C_basis.append(v)
-    C_basis = list(map(lambda x: x[3:], C_basis))
+    This construction is tested in ``generators_test.py``, where the
+    result is compared with the result from this method.
 
-    code = LinearCode(Matrix(GF(2), C_basis))
+    REFERENCES:
 
-    G = code.cosetGraph()
+    The description and construction of this graph can be found in
+    [BCN1989]_, page 365.
+    """
+    import lzma
+    from importlib.resources import as_file, files
+    from pickle import load
+
+    # Path to the pickled-and-xz'd list of (vertices, edges)
+    ppath = files('sage.graphs.generators').joinpath(
+      "shortened_000_111_extended_binary_Golay_code_graph.pickle.xz"
+    )
+
+    with as_file(ppath) as p:
+        with lzma.open(p) as f:
+            vs_and_es = load(f, fix_imports=False)
+
+    # Vertices/edges are pickled as tuples of ints, but should be
+    # vectors with entries in GF(2).
+    V = VectorSpace(GF(2), 21)
+    for i in range(2048):
+        # vertex i
+        vs_and_es[0][i] = V(vs_and_es[0][i])
+        vs_and_es[0][i].set_immutable()
+    for i in range(21504):
+        # edge i = (v1, v2, l)
+        vs_and_es[1][i][0] = V(vs_and_es[1][i][0]) # v1
+        vs_and_es[1][i][0].set_immutable()
+        vs_and_es[1][i][1] = V(vs_and_es[1][i][1]) # v2
+        vs_and_es[1][i][1].set_immutable()
+
+    G = Graph(vs_and_es, format='vertices_and_edges')
     G.name("Shortened 000 111 extended binary Golay code")
     return G
 
diff --git a/src/sage/graphs/generators/generators_test.py b/src/sage/graphs/generators/generators_test.py
new file mode 100644
index 00000000000..5b57f1cd459
--- /dev/null
+++ b/src/sage/graphs/generators/generators_test.py
@@ -0,0 +1,34 @@
+import pytest
+
+
+def test_shortened_000_111_extended_binary_Golay_code_graph():
+    r"""
+    Test that Sage produces a graph equal to the one that we get
+    from this construction.
+
+    The construction itself takes a long time.
+    """
+    from sage.coding import codes_catalog
+    from sage.coding.linear_code import LinearCode
+    from sage.graphs.generators.distance_regular import (
+      shortened_000_111_extended_binary_Golay_code_graph
+    )
+    from sage.matrix.constructor import matrix
+    from sage.rings.finite_rings.finite_field_constructor import FiniteField
+
+    code = codes_catalog.GolayCode(FiniteField(2))
+    C_basis = code.basis()
+
+    # now special shortening
+    v = C_basis[0] + C_basis[1] + C_basis[2]  # v has 111 at the start
+    C_basis = C_basis[3:]
+    C_basis.append(v)
+    C_basis = list(map(lambda x: x[3:], C_basis))
+
+    code = LinearCode(matrix(FiniteField(2), C_basis))
+    G = code.cosetGraph()
+    G.name("Shortened 000 111 extended binary Golay code")
+    assert G.is_distance_regular()
+
+    H = shortened_000_111_extended_binary_Golay_code_graph()
+    assert G == H
diff --git a/src/sage/graphs/generators/meson.build b/src/sage/graphs/generators/meson.build
index c98c2647c25..e086316e599 100644
--- a/src/sage/graphs/generators/meson.build
+++ b/src/sage/graphs/generators/meson.build
@@ -6,9 +6,11 @@ py.install_sources(
   'classical_geometries.py',
   'degree_sequence.py',
   'families.py',
+  'generators_test.py',
   'intersection.py',
   'platonic_solids.py',
   'random.py',
+  'shortened_000_111_extended_binary_Golay_code_graph.pickle.xz',
   'smallgraphs.py',
   'world_map.py',
   subdir: 'sage/graphs/generators',
diff --git a/src/sage/graphs/generators/shortened_000_111_extended_binary_Golay_code_graph.pickle.xz b/src/sage/graphs/generators/shortened_000_111_extended_binary_Golay_code_graph.pickle.xz
new file mode 100644
index 00000000000..e46e00615d7
Binary files /dev/null and b/src/sage/graphs/generators/shortened_000_111_extended_binary_Golay_code_graph.pickle.xz differ