This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid and, thus, generalizes invariants that lie at the heart of several fundamental problems in a coding theory. An upper bound on the covering dimension is proved, improving Kung's classical bound for the critical exponent. In addition, a construction is given for linear codes that attain equality in this covering dimension bound.

A formula of channel matrix for linear codes over Zmis presented. m-ary PSK coherent-states are used as letter states and the square-root measurement (SRM) is used as a decoding. First, arbitrary linear codes over Zmare shown to group covariant. Then the formula of channel matrix is derived by generalizing [T.S. Usuda, S. Usami, I. Takumi, and M. Hata, Phys. Lett. A305, pp.125-134, (2002)]. Furthermore, the usefulness of the formula is considered.

This paper presents new connections between designs and matroids. We generalize the Assmus–Mattson theorem and another coding-theoretical theorem with respect to matroids, and thereby present new sufficient conditions for obtaining t-designs from matroids. These conditions may be relaxed for some self-dual matroids. We use our results to prove new constructions of t-designs from linear codes, including several new extensions and variants of the Assmus–Mattson theorem. We also present weighted t-designs which generalize t-designs. New t-designs are obtained from our results.

An efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices. By this algorithm, the higher weight enumerators of the unique doubly-even, self-dual code are calculated. The algorithm is based on a previously shown relationship between Tutte polynomials and higher weight enumerators.

In this correspondence, we study bounds on the Euclidean, Hamming, Lee, and Bachoc weights of codes over rings of order 4 similar to the Singleton bound and investigate the relationship between these bounds. Moreover, we give some characterizations of the codes meeting these bounds.

In this correspondence, we study maximum distance with respect to rank (MDR) codes over the ring Z/sub k/. We generalize the construction of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon codes and apply the generalized Chinese remainder theorem to construct codes.