In this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8, providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far.

On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q / Timpanella, M.; Zini, G.. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - (2023), pp. 1-24. [10.1007/s10623-022-01179-0]

On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q

Zini G.
2023

Abstract

In this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8, providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far.
2023
1
24
On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q / Timpanella, M.; Zini, G.. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - (2023), pp. 1-24. [10.1007/s10623-022-01179-0]
Timpanella, M.; Zini, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1328610
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