Ternary maximal self-orthogonal codes of lengths 21,2221,22 and 2323

We give a classification of ternary maximal self-orthogonal codes of lengths $21,22$ and $23$. This completes a classification of ternary maximal self-orthogonal codes of lengths up to $24$

A gluing technique for constructing relatively self-dual codes

AbstractIn this paper, we introduce self-dual codes relative to certain symmetric bilinear forms over a finite commutative ring. By refining the gluing theory of Conway, Pless, and Sloane, we obtain a gluing technique for constructing relatively self-dual codes. As examples of application of our technique, we find a construction of a self-dual binary [2(m + 3), m + 3, 6]-code from a self-dual [2m, m, l]-code with l⩾6, and a construction of doubly-even binary self-dual [2(m + 4), m + 4, 8]-code from a doubly even self-dual [2m, m, t]-code with t ⩾ 8

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