Extending two families of maximum rank distance codes. (English) Zbl 1500.94083
This paper deals with the construction of MRD codes. In particular, the authors provide a new family of \(2\)-dimensional \(\mathbb{F}_{q^n}\)-linear MRD codes that properly contains the two families found by G. Longobardi and C. Zanella [J. Algebr. Comb. 53, No. 3, 639–661 (2021; Zbl 1465.94113)] and by G. Longobardi et al. [“A large family of maximum scattered linear sets of \(\mathrm{PG}(1,qn)\) and their associated MRD codes”, Preprint, arXiv:2102.08287]. The crucial part is proving that the constructed codes are in fact MRD. Since the authors focus on square matrices, they can identify the \(\mathbb{F}_q\)-algebra of \(n\times n\) matrices as the algebra of \(\sigma\)-polynomials, i.e. polynomials of the form
\[
f(x)=\sum_{i=0}^{n-1} f_ix^{\sigma^i}, \quad f_i\in\mathbb{F}_{q^n},
\]
where \(\sigma\) is a generator of the Galois group \(\mathrm{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)\), and \(x^{\sigma}\) denotes the action of \(\sigma\) on \(x\). The authors give a general argument which allows to extend any construction of MRD codes based on \(\sigma\)-polynomials to any other generator \(\theta\) of the Galois group \(\mathrm{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)\) under certain assumptions. Furthermore, they are able to show that the new codes they introduced are inequivalent to all the other known \(\mathbb{F}_{q^n}\)-linear MRD codes. Moreover, they investigate when two codes in this new wider family are equivalent, which turns to provide a lower bound on the number of equivalence classes of codes in this new family.
Reviewer: Gianira N. Alfarano (Zürich)
MSC:
| 94B05 | Linear codes (general theory) |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 11T06 | Polynomials over finite fields |
| 51E22 | Linear codes and caps in Galois spaces |
Citations:
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