Directed graph representation of half-rate additive codes over GF(4)

Abstract

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.

Vector-Circulant Matrices and Vector-Circulant Based Additive Codes over Finite Fields

“…Since additive codes over finite fields have applications in both classical and quantum communications (see, for example, [1][2][3][4][5]), it is of natural interest to study this family of codes. Constructions of good/optimal additive codes have been widely studied (see [3,[8][9][10], and references therein).…”

“…Characterizations of self-dual and formally self-dual additive codes have been given in [11,12], respectively. Circulant based additive codes and cyclic additive codes have been studied in [1,8], respectively. Here, we focus on the construction of additive codes based on vector-circulant matrices.…”

“…Coding theory has been introduced to deal with this problem since the 1960s. Additive codes constitute an important class of codes due to their rich algebraic structures and wide applications in both classical and quantum communications (see [1][2][3][4][5], and references therein).…”

“…We note that the codes of length 2 to 7 are extremal and the codes of length 8 to 13 are near-extremal. Comparing Table 5 in [1], and Table 1 in [9], the codes given in Table 1 are optimal. Table 1.…”

The nonexistence of an additive quaternary [15,5,9]-code

“…In [4,6] we determine the optimal parameters for all lengths n ≤ 13 except in one case. The last gap was closed by Danielsen and Parker [8] who constructed two cyclic [13, 6.5, 6]-codes. Let us concentrate on lengths n = 14 and n = 15 now.…”