Directed Graph Representation of Half-Rate Additive Codes over GF(4)

We show that (n,2^n) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n,2^n) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.

💡 Research Summary

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This paper establishes a one‑to‑one correspondence between half‑rate additive codes over GF(4) – that is, codes of length n with 2ⁿ codewords – and directed graphs on n vertices. The correspondence generalises the well‑known undirected‑graph representation of self‑dual additive codes. For a directed graph G with adjacency matrix A (entries 0 or 1), the authors define a generator matrix

  G = Iₙ + ωA,

where ω is a primitive element of GF(4) (ω² = ω + 1). The rows of this matrix form a basis of an (n, 2ⁿ) additive code. Crucially, column operations by GL(2, GF(4)) and column permutations on the code correspond respectively to edge‑reversal and vertex‑permutation on the graph. Hence two codes are equivalent if and only if their associated directed graphs are isomorphic. This graph‑based viewpoint dramatically reduces the computational burden of code classification because graph isomorphism can be tested far more efficiently than exhaustive matrix equivalence checks.

Using this framework, the authors exhaustively enumerate all non‑isomorphic directed graphs for n ≤ 7 and construct the corresponding additive codes. They compute minimum distances, weight enumerators, and dual relationships, thereby obtaining a complete catalogue of (n, 2ⁿ) codes up to length 7, including the subclasses of isodual (code equivalent to its dual) and formally self‑dual (code whose weight enumerator coincides with that of its dual) codes. The catalogue reveals many codes that are not self‑dual yet possess the same weight distribution as their duals, expanding the known landscape beyond the previously studied self‑dual family.

The paper then introduces two families of structured directed‑graph codes: circulant and bordered‑circulant constructions. In a circulant directed graph the adjacency matrix is a binary circulant matrix; the bordered version adds an extra row and column to enforce additional symmetry. The authors prove that any code derived from either construction is automatically isodual, because the underlying graph possesses a vertex‑permutation that maps the graph to its transpose. This theoretical result supplies a large, easily generated pool of high‑quality codes.

A systematic computer search was performed on these families for lengths up to 26. The search employed nauty for graph isomorphism, GAP/Magma for algebraic verification, and exhaustive distance calculations. The most striking findings are:

• For lengths 11 and 13 the authors discovered formally self‑dual codes whose minimum distances exceed the best previously known formally self‑dual codes by one, achieving near‑extremal parameters.
• For lengths 24, 25, and 26 they produced isodual codes with minimum distances 9, 9, and 10 respectively, surpassing the best known self‑dual codes (which have distances 8, 8, and 9). Thus the directed‑graph approach yields codes that are strictly better than the optimal self‑dual counterparts at these lengths.

These results demonstrate that allowing directed edges – i.e., abandoning the symmetry constraint of undirected graphs – opens up a richer design space, enabling higher minimum distances while preserving desirable duality properties. The authors also discuss implications for quantum error‑correcting codes, noting that additive GF(4) codes correspond to stabilizer codes; the newly found isodual and formally self‑dual codes can be directly translated into quantum codes with improved distance.

The paper concludes with several avenues for future work: extending the directed‑graph representation to longer lengths, exploring analogous constructions over larger finite fields such as GF(8), and developing algorithmic methods to automatically optimise directed‑graph parameters for targeted code properties. Overall, the work bridges additive coding theory and graph theory, providing both a powerful theoretical tool for classification and a practical source of high‑performance codes.