Some Hadamard matrices of order 32 and their binary codes

It is known that all doubly‐even self‐dual codes of lengths 8 or 16, and the extended Golay code, can be constructed from some binary Hadamard matrix of orders 8, 16, and 24, respectively. In this note, we demonstrate that every extremal doubly‐even self‐dual [32,16,8] code can be constructed from some binary Hadamard matrix of order 32. © 2004 Wiley Periodicals, Inc.     

On binary codes for identification

A code C ⊆ 𝔽₂ⁿ is called t‐identifying if the sets B_t(x) ∩ C are all nonempty and different. Constructions of t‐identifying codes are given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 151–156, 2000     

On a complete decoding scheme for binary radical codes

This work was partially supported by the Fulbright commission, while the author was working at the Princeton University, Princeton NJ 08544, USA.     

Bounds for binary multiple covering codes

We study codes that are multiple coverings of the Hamming space and discuss lower and upper bounds onK(n, r, ), the minimum cardinality of a binary code of lengthn such that the Hamming spheres of radiusr centered at the codewords cover at least times. We also study the similar problem of multiple coverings containing repeated words. A table of bounds forn16,r4, 4 is given.     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

Evaluation codes and plane valuations

We apply tools coming from singularity theory, as Hamburger–Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp–Massey–Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.     

Classification of binary covering codes

The minimum size of a binary covering code of length n and covering radius r is denoted by K (n, r) and corresponding codes are called optimal. In this article a classification up to equivalence of all optimal covering codes having either length at most 8 or cardinality at most 4 is completed. Moreover, we prove that K (9, 2) = 16. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 391–401, 2000     

Combinatorial lemma and structure of a class of binary codes

A class of complete binary code systems is presented in which one of the symbols of the alphabet is used in any code combination not more than once, in terms of prefix codes of this class and of the operations of inversion and substitution of codes.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 325–331, March, 1970.In conclusion the author expresses his gratitude to G. Pollak who proposed to characterize the factorizations N by a system of calculus with a variable base, and to L. Redei who discussed the problems related to the factorization of finite sets.     

Locality of optimal binary codes

The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with $r=1,2\text{ and }3$. Based on these results, for $2\le k\le 7$ and $n\ge k+2$, we give an optimal linear $[n,k,d]$ code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for $n<2(2^k-1)$ and by the Griesmer bound for $n\ge 2(2^k-1)$. Almost all the $[n,k,d]$ codes ($2\le k\le 7$) have locality $r\le 3$ except for the three codes, and most of the $[n,k,d]$ code with $n<2(2^k-1)$ achieves the Cadambe–Mazumdar bound for LRCs.     

Varieties of binary linear codes

Received October 9, 1998; accepted in final form May 13, 1999.     

Construction of binary LCD codes,ternary LCD codes and quaternary Hermitian LCD codes

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bounds on the largest minimum weights.

     

Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three

Ternary self-orthogonal codes with dual distance three and ternary quantum codes of distance three constructed from ternary self-orthogonal codes are discussed in this paper. Firstly, for given code length n ≥ 8, a ternary [n, k]₃ self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n ≥ 8, two nested ternary self-orthogonal codes with dual distance two and three are constructed, and consequently ternary quantum code of length n and distance three is constructed via Steane construction. Almost all of these quantum codes constructed via Steane construction are optimal or near optimal, and some of these quantum codes are better than those known before.     

Permutation decoding for binary codes from lattice graphs

By finding explicit PD sets, we show that permutation decoding can be used for the binary code obtained from the row span over the field F₂ of an adjacency matrix of the lattice graph L₂(n) for any n?5.     

On the [28, 7, 12] binary self-complementary codes and their residuals

Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA ₀=A ₂₈=1,A ₁₂=A ₁₆=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.This work has been done during the visit of the author as a guest researcher to the Department of Electrical Engineering at Linköping University, Sweden. The research was partially supported by the Bulgarian National Science Foundation under contracts No. 37/1987 and No. 876/1988.