A search algorithm for linear codes: progressive dimension growth

Tsvetan Asamov, Nuh Aydin
<span title="2007-08-10">2007</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
This paper presents an algorithm, called progressive dimension growth (PDG), for the construction of linear codes with a pre-specified length and a minimum distance. A number of new linear codes over G F(5) that have been discovered via this algorithm are also presented. AMS Classification 94B05 Introduction A linear code over G F(q), the finite field with q elements, of length n, dimension k, and minimum distance (weight) d is referred to as an [n, k, d] q -code. One of the most important and
more &raquo; ... hallenging problems in coding theory is to find optimal values of the parameters of a linear code and to explicitly construct codes whose parameters attain, or get as close as possible to, those optimal values. One formulation of the problem is: Given a q-ary linear code C with parameters [n, k], find d q [n, k], the largest value of the minimum distance. Another formulation would be: Given a length n, and a minimum distance d, what is the largest value of k, the dimension of the code? We denote this value by k q [n, d]. Similarly, one can fix the dimension and the minimum distance and ask for the smallest possible value of n.
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