Torleiv Kløve

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

Abstract We study the weight distribution of irreducible cyclic ( n , k ) codeswith block lengths n = n 1 (( q 1 − 1)/ N ), where N | q − 1, gcd ( n 1 , N ) = 1, and gcd ( l , N ) = 1. We present the weight enumerator polynomial, A ( z ), when k = n 1 l , k = ( n 1 − 1) l , and k = 2 l . We also show how to find A ( z ) in general by studying the generator matrix of an ( n 1 , m ) linear code, V ∗ d over GF ( q d ) where d = gcd ( ord n 1 ( q ), l ). Specifically we study A ( z ) when V ∗ d is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A μ which completely determine the weight distributionof any irreducible cyclic ( n 1 (2 1 − 1), k ) code over GF(2) for all n 1 ⩽ 17.

Generalized Hamming weights of linear codes

The generalized Hamming weight, d/sub r/(C), of a binary linear code C is the size of the smallest support of any r-dimensional subcode of C. The parameter d/sub r/(C) determines the codes performance on the wire-tap channel of Type II. Bounds on d/sub r/(C), and in some cases exact expressions, are derived. In particular, a generalized Griesmer bound for d/sub r/(C) is presented and examples are given of codes meeting this bound with equality. >

Permutation arrays for powerline communication and mutually orthogonal latin squares

We develop a connection between permutation arrays that are used in powerline communication and well-studied combinatorial objects, mutually orthogonal latin squares (MOLS). From this connection, many new results on permutation arrays can be obtained.

Support weight distribution of linear codes

Abstract The main result of the paper is expressions for the support weight distributions of a linear code in terms of the support weight distributions of the dual code.

Codes for Error Detection

There are two basic methods of error control for communication, both involving coding of the messages. With forward error correction, the codes are used to detect and correct errors. In a repeat request system, the codes are used to detect errors and, if there are errors, request a retransmission. Error detection is usually much simpler to implement than error correction and is widely used. However, it is given a very cursory treatment in almost all textbooks on coding theory. Only a few older books are devoted to error detecting codes. This book begins with a short introduction to the theory of block codes with emphasis on the parts important for error detection. The weight distribution is particularly important for this application and is treated in more detail than in most books on error correction. A detailed account of the known results on the probability of undetected error on the q-ary symmetric channel is also given.

On the covering radius of binary codes (Corresp.)

Upper bounds on the covering radius of binary codes are studied. In particular it is shown that the covering radius r_{m} of the first-order Reed-Muller code of lenglh 2^{m} satisfies 2^{m-l}-2^{\lceil m/2 \rceil -1} r_{m} \leq 2^{m-1}-2^{m/2-1} .

Meeting the Welch and Karystinos-Pados Bounds on DS-CDMA Binary Signature Sets

The Welch lower bound on the total-squared-correlation (TSC) of binary signature sets is loose for binary signature sets whose length L is not a multiple of 4. Recently Karystinos and Pados [6,7] developed new bounds that are better than the Welch bound in those cases, and showed how to achieve the bounds with modified Hadamard matrices except in a couple of cases. In this paper, we study the open cases.

Distance-preserving mappings from binary vectors to permutations

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes. In this article, several explicit constructions of such mappings preserving or increasing the Hamming distance are given. Some applications are given to illustrate the usefulness of the construction. In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given.

Systematic, Single Limited Magnitude Error Correcting Codes for Flash Memories

A relatively new model of error correction is the limited magnitude error model. That is, it is assumed that the absolute difference between the sent and received symbols is bounded above by a certain value l. In this paper, we propose systematic codes for asymmetric limited magnitude channels that are able to correct a single error. We also show how this construction can be slightly modified to design codes that can correct a single symmetric error of limited magnitude. The designed codes achieve higher code rates than single error correcting codes previously given in the literature.

Constructions of permutation arrays

A permutation array (PA) of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n - d positions. Some constructions of PAs are given.