Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound

A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois

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