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

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

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201