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Hamming weight distributions of multi-twisted codes over finite fields

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Abstract

Let \({\mathbb {F}}_q\) denote the finite field of order q,  and let \(n = m_1+m_2+\cdots +m_\ell ,\) where \(m_1,m_2,\ldots ,m_\ell \) are arbitrary positive integers (not necessarily coprime to q). In this paper, we explicitly determine Hamming weights of all non-zero codewords of several classes of multi-twisted codes of length n and block lengths \((m_1,m_2,\ldots ,m_\ell )\) over \({\mathbb {F}}_q.\) As an application of these results, we explicitly determine Hamming weight distributions of several classes of multi-twisted codes of length n and block lengths \((m_1,m_2,\ldots , m_{\ell })\) over \({\mathbb {F}}_q.\) Among these classes of multi-twisted codes, we identify two classes of optimal equidistant linear codes that have nice connections with the theory of combinatorial designs and several other classes of minimal linear codes that are useful in constructing secret sharing schemes with nice access structures. We illustrate our results with some examples, and list many optimal, projective and minimal linear codes belonging to these classes of multi-twisted codes.

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Authors and Affiliations

  1. Department of Mathematics, IIIT-Delhi, New Delhi, 110020, India

    Varsha Chauhan, Anuradha Sharma, Sandeep Sharma & Monika Yadav

Authors
  1. Varsha Chauhan
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  2. Anuradha Sharma
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  3. Sandeep Sharma
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  4. Monika Yadav
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Correspondence to Anuradha Sharma.

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Communicated by C. Ding.

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V. Chauhan: Research support by UGC, India, is gratefully acknowledged. A. Sharma: Research support by DST-SERB, India, under Grant No. MTR/2017/000358 is gratefully acknowledged. S. Sharma: Research support by UGC, India, is gratefully acknowledged, M. Yadav: Research support by CSIR, India, is gratefully acknowledged.

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Chauhan, V., Sharma, A., Sharma, S. et al. Hamming weight distributions of multi-twisted codes over finite fields. Des. Codes Cryptogr. 89, 1787–1837 (2021). https://doi.org/10.1007/s10623-021-00889-1

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  • DOI: https://doi.org/10.1007/s10623-021-00889-1

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