Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Self-Dual and Complementary Dual Abelian Codes over Galois Rings

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${\rm GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${\rm GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${\rm GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${\rm GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${\rm GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${\rm GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${\rm GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.Comment: 22 page

Graph-Based Classification of Self-Dual Additive Codes over Finite Fields

Quantum stabilizer states over GF(m) can be represented as self-dual additive codes over GF(m^2). These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over GF(4). In this paper we classify self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over GF(9) by using an extension technique. We prove that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure

New extremal singly even self-dual codes of lengths 6464 and 6666

For lengths $64$ and $66$, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169