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On Codes with Distances \(d\) and \(n\)

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Abstract

We enumerate all \(q\)-ary additive (in particular, linear) block codes of length \(n\) and cardinality \(N\ge q^2\) with exactly two distances: \(d\) and \(n\). For arbitrary codes of length \(n\) with distances \(d\) and \(n\), we obtain upper bounds on the cardinality via linear programming and using relationships to 2-distance sets on a Euclidean sphere.

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Funding

The research of P. Boyvalenkov was supported in part by the Bulgarian National Science Foundation, project no. KP-06-Russia/33-2020.

The research of K. Delchev was supported in part by the Bulgarian National Science Foundation, project no. KP-06-N32/2-2019.

The research of V.A. Zinoviev and D.V. Zinoviev was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences within the program of fundamental research on the topic “Mathematical Foundations of the Theory of Error-Correcting Codes” and was also supported by the National Science Foundation of Bulgaria under project no. 20-51-18002.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    P. Boyvalenkov & K. Delchev

  2. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    V. A. Zinoviev & D. V. Zinoviev

Authors
  1. P. Boyvalenkov
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  2. K. Delchev
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  3. V. A. Zinoviev
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  4. D. V. Zinoviev
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Additional information

Translated from Problemy Peredachi Informatsii, 2022, Vol. 58, No. 4, pp. 62–83. https://doi.org/10.31857/S0555292322040064

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Boyvalenkov, P., Delchev, K., Zinoviev, V. et al. On Codes with Distances \(d\) and \(n\). Probl Inf Transm 58, 352–371 (2022). https://doi.org/10.1134/S0032946022040068

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