THE CONCATENATED STRUCTURE OF QUASI-ABELIAN CODES

Martino Borello (1) , Cem Güneri (2) , Elif Saçikara (3) , Patrick Solé (4)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Sabanci University [Istanbul]
3 UZH - Universität Zürich [Zürich] = University of Zurich
4 I2M - Institut de Mathématiques de Marseille

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in (Jitman, Ling, (2015)), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling, Solé, (2001)). We give a concatenated decomposition of quasi-abelian codes and show, as in the quasi-cyclic case, that the two decompositions are equivalent. The concatenated decomposition allows us to give a general minimum distance bound for quasi-abelian codes and to construct some optimal codes. Moreover, we show by examples that the minimum distance bound is sharp in some cases. In addition, examples of large strictly quasi-abelian codes of about a half rate are given. The concatenated structure also enables us to conclude that strictly quasi-abelian linear complementary dual codes over any finite field are asymptotically good.

Keywords: Quasi-abelian codes; concatenated codes; linear complementary dual codes; optimal codes; additive abelian codes

Domains: Information Theory [cs.IT]

Martino Borello, Cem Güneri, Elif Saçikara, Patrick Solé. THE CONCATENATED STRUCTURE OF QUASI-ABELIAN CODES. Designs, Codes and Cryptography, 2021, ⟨10.1007/s10623-021-00921-4⟩. ⟨hal-03346416⟩