Paper 2023/1725

Few-weight linear codes over $\mathbb{F}_p$ from $t$-to-one mappings

René Rodríguez-Aldama, University of Primorska
Abstract

For any prime number $p$, we provide two classes of linear codes with few weights over a $p$-ary alphabet. These codes are based on a well-known generic construction (the defining-set method), stemming on a class of monomials and a class of trinomials over finite fields. The considered monomials are Dembowski-Ostrom monomials $x^{p^{\alpha}+1}$, for a suitable choice of the exponent $\alpha$, so that, when $p>2$ and $n\not\equiv 0 \pmod{4}$, these monomials are planar. We study the properties of such monomials in detail for each integer $n$ greater than two and any prime number $p$. In particular, we show that they are $t$-to-one, where the parameter $t$ depends on the field $\mathbb{F}_{p^n}$ and it takes the values $1, 2$ or $p+1$. Moreover, we give a simple proof of the fact that the functions are $\delta$-uniform with $\delta \in \{1,4,p\}$. This result describes the differential behaviour of these monomials for any $p$ and $n$. For the second class of functions, we consider an affine equivalent trinomial to $x^{p^{\alpha}+1}$, namely, $x^{p^{\alpha}+1}+\lambda x^{p^{\alpha}}+\lambda^{p^{\alpha}}x$ for $\lambda\in \mathbb{F}_{p^n}^*$. We prove that these trinomials satisfy certain regularity properties, which are useful for the specification of linear codes with three or four weights that are different than the monomial construction. These families of codes contain projective codes and optimal codes (with respect to the Griesmer bound). Remarkably, they contain infinite families of self-orthogonal and minimal $p$-ary linear codes for every prime number $p$. Our findings highlight the utility of studying affine equivalent functions, which is often overlooked in this context.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Preprint.
Keywords
Finite fields$p$-ary functionslinear codesminimal codesoptimal codesself-orthogonal codes
Contact author(s)
rene7ca @ gmail com
History
2023-11-13: approved
2023-11-07: received
See all versions
Short URL
https://ia.cr/2023/1725
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2023/1725,
      author = {René Rodríguez-Aldama},
      title = {Few-weight linear codes over $\mathbb{F}_p$ from $t$-to-one mappings},
      howpublished = {Cryptology {ePrint} Archive, Paper 2023/1725},
      year = {2023},
      url = {https://eprint.iacr.org/2023/1725}
}
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