Keisuke Shiromoto

Maximum distance codes over rings of order 4

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.

MDR codes over Z/sub k/

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.

Wei-type duality theorems for matroids

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in each case yielding a duality theorem for the respective class of objects.

Singleton Bounds for Codes over Finite Rings

We introduce the Singleton bounds for codes over a finite commutative quasi-Frobenius ring.

A basic exact sequence for the Lee and Euclidean weights of linear codes over Zℓ

Abstract Recently, many papers have been published dealing with codes over finite rings. In this paper, we introduce some Singleton type bounds of Lee and Euclidean weights for codes over the ring Z l and give a proof of the MacWilliams identities for the Lee and Euclidean weight enumerators.

A construction of mutually disjoint Steiner systems from isomorphic Golay codes

It is well known that the extended binary Golay [24,12,8] code yields 5-designs. In particular, the supports of all the weight 8 codewords in the code form a Steiner system S(5,8,24). In this paper, we give a construction of mutually disjoint Steiner systems S(5,8,24) by constructing isomorphic Golay codes. As a consequence, we show that there exists at least 22 mutually disjoint Steiner systems S(5,8,24). Finally, we prove that there exists at least 46 mutually disjoint 5-(48,12,8) designs from the extended binary quadratic residue [48,24,12] code.

A MacWilliams type identity for matroids

In this paper, we consider the coboundary polynomial for a matroid as a generalization of the weight enumerator of a linear code. By describing properties of this polynomial and of a more general polynomial, we investigate the matroid analogue of the MacWilliams identity. From coding-theoretical approaches, upper bounds are given on the size of circuits and cocircuits of a matroid, which generalizes bounds on minimum Hamming weights of linear codes due to I. Duursma.

The Higher Weight Enumerators of the Doubly-Even, Self-Dual

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.

[48, 24, 12]

We have the relationships between the Hamming weight enumerator of linear codes over GFqm which have generator matrices over GFq, the support weight enumerator and the λ-ply weight enumerator.

Code

We establish a bound on the minimum ρ distance for codes in Matn,s (Zk) with respect to their ranks and call codes meeting this bound MDR codes. We extend the relationship between codes in Matn,s (Zk) and distributions in the unit cube and use the Chinese Remainder Theorem to construct codes and distributions.