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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

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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