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Octanary linear codes using simplicial complexes

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Abstract

Certain simplicial complexes are used to construct a subset D of \(\mathbb {F}_{2^{n}}^{m}\) and D, in turn, defines the linear code CD over \(\mathbb {F}_{2^{n}}\) that consists of (v ⋅ d)d∈D for \(v\in \mathbb {F}_{2^{n}}^{m}\). Here we deal with the case n = 3, that is, when CD is an octanary code. We establish a relation between CD and its binary subfield code \(C_{D}^{(2)}\) with the help of a generator matrix. For a given length and dimension, a code is called distance optimal if it has the highest possible distance. With respect to the Griesmer bound, three infinite families of distance optimal codes are obtained, and sufficient conditions for certain linear codes to be minimal are established.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The authors also would like to thank the handling editor of the journal for the time spent during the whole process.

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  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, Delhi, India

    Vidya Sagar & Ritumoni Sarma

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  1. Vidya Sagar
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  2. Ritumoni Sarma
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Vidya Sagar and Ritumoni Sarma contributed equally to this work. Both authors read and approved the final manuscript.

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Correspondence to Vidya Sagar.

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Sagar, V., Sarma, R. Octanary linear codes using simplicial complexes. Cryptogr. Commun. 15, 599–616 (2023). https://doi.org/10.1007/s12095-022-00617-z

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  • DOI: https://doi.org/10.1007/s12095-022-00617-z

Keywords

Mathematics Subject Classification (2020)