Jon-Lark Kim

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known \(q\) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters.

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called {\em{duadic double circulant codes}}, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb F_2, \mathbb F_3, \mathbb F_4, \mathbb F_5$, and $\mathbb F_7$. In particular, we find a new ternary self-dual $[76,38,18]$ code and easily rediscover optimal binary self-dual codes with parameters $[66,33,12]$, $[68,34,12]$, $[86,43,16]$, and $[88,44,16]$ as well as a formally self-dual binary $[82,41,14]$ code.

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 $\mathbb F_4$.

We complete the building-up construction for self-dual codes by resolving the open cases over $GF(q)$ with $q \equiv 3 \pmod 4$, and over $\Z_{p^m}$ and Galois rings $\GR(p^m,r)$ with an odd prime $p$ satisfying $p \equiv 3 \pmod 4$ with $r$ odd. We also extend the building-up construction for self-dual codes to finite chain rings. Our building-up construction produces many new interesting self-dual codes. In particular, we construct 945 new extremal self-dual ternary $[32,16,9]$ codes, each of which has a trivial automorphism group. We also obtain many new self-dual codes over $\mathbb Z_9$ of lengths $12, 16, 20$ all with minimum Hamming weight 6, which is the best possible minimum Hamming weight that free self-dual codes over $\Z_9$ of these lengths can attain. From the constructed codes over $\mathbb Z_9$, we reconstruct optimal Type I lattices of dimensions $12, 16, 20,$ and 24 using Construction $A$; this shows that our building-up construction can make a good contribution for finding optimal Type I lattices as well as self-dual codes. We also find new optimal self-dual $[16,8,7]$ codes over GF(7) and new self-dual codes over GF(7) with the best known parameters $[24,12,9]$.

In this paper we classify all extremal and $s$-extremal binary self-dual codes of length 38. There are exactly 2744 extremal $[38,19,8]$ self-dual codes, two $s$-extremal $[38,19,6]$ codes, and 1730 $s$-extremal $[38,19,8]$ codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of $s$-extremal $[38,19,6]$ codes permits to achieve the classification of all $s$-extremal codes with d=6.