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New linear codes over non-prime fields

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Abstract

One of the most important and challenging problems in coding theory is to explicitly construct linear codes with best possible parameters. Computers are often used to search for optimal codes. However, given the large size of the search space and computational complexity of determining the minimum distance, researchers usually focus on promising classes of linear codes with rich algebraic structures. One such class of codes is quasi-twisted (QT) codes which contains cyclic, constacyclic, and quasi-cyclic (QC) codes as sub-classes. Improving and automatizing existing search algorithms for QT codes, we have been able to obtain 64 record-breaking linear codes (codes with better parameters than currently best known linear codes) over the non-prime fields \(\mathbb {F}_{4}\), \(\mathbb {F}_{8}\) and \(\mathbb {F}_{9}\). Moreover, we obtained 82 additional new codes from the standard constructions of puncturing, extending and shortening a code. Further, we have found 15 QT codes that are new among the class of QT codes.

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Notes

  1. Note that when we use the standard generator for a constacyclic code, the shift constant does not show up in the first coordinate of each row vector because they are all zeros. If we use a generator of a higher degree however, then it will be necessary to include the shift constant a.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Kenyon College, Gambier, OH, 43022, USA

    Nuh Aydin, Ghada Bakbouk & Jonathan G. G. Lambrinos

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  1. Nuh Aydin
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  2. Ghada Bakbouk
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  3. Jonathan G. G. Lambrinos
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Correspondence to Nuh Aydin.

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This work was supported by Kenyon College Summer Science Scholars program.

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Aydin, N., Bakbouk, G. & Lambrinos, J.G.G. New linear codes over non-prime fields. Cryptogr. Commun. 11, 891–902 (2019). https://doi.org/10.1007/s12095-018-0333-2

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