New linear codes with prescribed group of automorphisms found by heuristic search

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May 2009, 3(2): 157-166. doi: 10.3934/amc.2009.3.157

New linear codes with prescribed group of automorphisms found by heuristic search

Axel Kohnert ^1, and Johannes Zwanzger ^1,

1.	Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany, Germany

Received December 2008 Revised March 2009 Published May 2009

Abstract
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In this paper, we present a new heuristic algorithm for solving certain systems of Diophantine inequalities. A variant which involves Monte-Carlo search is also applyable to more general problems. Our goal was the construction of point sets in PG$(k-1,q)$ with fixed cardinality and small maximal intersection number with the lines. These points sets correspond to $k$-dimensional linear codes over $\mathbb F_q$ with high minimum distance. We obtained them by prescribing a certain nontrivial subgroup of GL$(k,q)$ to be contained in their automorphism group. Following a method which was first introduced by Kramer and Mesner in the 1970s, this allows a strong reduction in the size of the corresponding Diophantine systems. Doing so we found a lot of new record breaking linear codes for the cases $q = 2, 3, 4, 5, 7, 8, 9$ from which at least $6$ are optimal.

Keywords: high minimum distance., Heuristic algorithm, linear codes, coding theory, automorphism group, Monte-Carlo.

Mathematics Subject Classification: Primary: 94B05; Secondary: 68T2.

Citation: Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157

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