Previously, (linear) codes over Z/sub 4/ and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study Z/sub 4/-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to Z/sub 4/ produces a new binary code, a (92, 2/sup 24/, 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.

One of the most important problems of coding theory is to construct codes with the best possible minimum distance. We further generalize the method first introduced by Gulliver and Harada (see Des., Codes Cryptogr, vol. 22, no. 1, p.89-96, 2001) and later generalized by the present authors, and obtain new linear codes which improve the best known minimum-distance bounds of certain linear codes. We have found eight new linear codes over F/sub 5/ with improved minimum distances. We introduce a generalized version of a Gray map, then we give definitions of quasi- and nearly quasi-cyclic codes. We conclude by giving the parameters of new linear codes with their generator matrices.

Berger and Charpin (see ibid., vol.42, p.2194-2209, 1996 and Des., Codes Cuyptogr., vol.18, no.1/3, p.29-53, 1999) devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. We extend this method to cyclic and extended cyclic codes over the Galois ring GR (p/sup a/, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2/sup m/ over Z/sub 4/, thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov. We give a large class of codes over Z/sub 4/ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes.

One of the most important problems of coding theory is to construct codes with best possible minimum distances. In this article, using the structure of quasi-cyclic (QC) codes with a BCH-type bound given by Lally and Fitzpatrick (see WCC '99, Workshop on Coding and Cryptography, Paris, France, 1999), we have found 20 new ternary quasi-cyclic codes with improved minimum distances.

The basic theory of t-UEC d-UED codes is developed. Methods for construction of such codes from symmetric error-correcting and asymmetric error-correcting codes are developed. Some bounds for t-EC d-UED codes are improved. Encoding/decoding procedures for these codes are discussed.>

It is shown that for $v $ sufficiently large and $k\geqq 2t + 1$, for any feasible quadruple $t - ( v ,k,\lambda )$ there exists a $t - ( v ,k,\lambda )$-design in which multiplicity of every block is 0 or $ \pm 1$ and the number of blocks with nonzero multiplicity is not too large compared to $\lambda \begin{pmatrix} v \\ t \end{pmatrix} $. As a consequence it is shown that the usual $t - ( v ,k,\lambda )$-designs in which no block is repeated more than twice exist if \[ \begin{pmatrix} v - t \\ k - t \end{pmatrix} + c_1 ( t,k )v ^{k - 2t} \geqq \lambda \geqq \begin{pmatrix} v - t \\ k - t \end{pmatrix} - c_1 ( t,k )v ^{k - 2t} \] where $c_1 ( t,k )$ is some function of t and k only. This implies that in Wilson’s result on the existence of a $t - ( v ,k,\lambda )$-design for \[ \lambda = m \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix} + \mu ,\quad 0\leqq \mu < \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix}, \] and m sufficiently large, the condition sufficiently large m can be replaced by $m\geqq 0$ when $\mu \geqq \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix} - c_1 ( t,k )v^{k - 2t} $ and by $m\geqq 1$ when $\mu \leqq c_1 ( t,k )v ^{k - 2t} $.