Computation of Minimum Hamming Weight for Linear Codes

Linear Binary Error Correcting Codes with Specified Minimum Hamming Distance

Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field Fq is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight dr (C), 1 r k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d1(C), d2(C), . . . , dk(C)) is called the Ha...

Weight Distributions of Hamming Codes

We derive a recursive formula determining the weight distribution of the [n = (qm − 1)/(q − 1), n − m, 3] Hamming code H(m, q), when (m, q−1) = 1. Here q is a prime power. The proof is based on Moisio’s idea of using Pless power moment identity together with exponential sum techniques.

Algorithms for the minimum weight of linear codes

We outline the algorithm for computing the minimum weight of a linear code over a finite field that was invented by A. Brouwer and later extended by K.-H. Zimmermann. We show that matroid partitioning algorithms can be used to efficiently find a favourable (and sometimes best possible) sequence of information sets on which the Brouwer-Zimmermann algorithm operates. We present a new algorithm fo...

Generalized Hamming weights for linear codes

Error control codes are widely used to increase the reliability of transmission of information over various forms of communications channels. The Hamming weight of a codeword is the number of nonzero entries in the word; the weights of the words in a linear code determine the error-correcting capacity of the code. The rth generalized Hamming weight for a linear code C, denoted by dr(C), is the ...

Weight Distributions of Hamming Codes (II)

In a previous paper, we derived a recursive formula determining the weight distributions of the [n = (qm − 1)/(q − 1), n−m, 3] Hamming code H(m,q), when (m, q−1) = 1. Here q is a prime power. We note here that the formula actually holds for any positive integer m and any prime power q, without the restriction (m,q − 1) = 1.