Designs, Codes and Cryptography

, Volume 70, Issue 1–2, pp 127–138 | Cite as

Classification and nonexistence results for linear codes with prescribed minimum distances

Article

Abstract

Starting from a linear [n, k, d] q code with dual distance \({d^{\bot}}\) , we may construct an \({[n - d^\bot, k - d^\bot +1,\geq d]_q}\) code with dual distance at least \({\left\lceil\frac{d^\bot}{q}\right\rceil}\) using construction Y 1. The inverse construction gives a rule for the classification of all [n, k, d] q codes with dual distance \({d^{\bot}}\) by adding \({d^\bot}\) further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] q , q =  2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.

Keywords

Classification Code equivalence Construction Y1 Linear code Residual code Semilinear isometry 

Mathematics Subject Classification

94B65 94B05 05E18 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany

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