Motivated by applications to noncoherent network coding, we study subspace codes defined by sets of linear cellular automata (CA). As a first remark, we show that a family of linear CA where the local rules have the same diameter—and thus the associated polynomials have the same degree—induces a Grassmannian code. Then, we prove that the minimum distance of such a code is determined by the maximum degree occurring among the pairwise greatest common divisors (GCD) of the polynomials in the family. Finally, we consider the setting where all such polynomials have the same GCD, and determine the cardinality of the corresponding Grassmannian code. As a particular case, we show that if all polynomials in the family are pairwise coprime, the resulting Grassmannian code has the highest minimum distance possible.
Mariot, L., Mazzone, F. (2023). On the Minimum Distance of Subspace Codes Generated by Linear Cellular Automata. In Cellular Automata and Discrete Complex Systems 29th IFIP WG 1.5 International Workshop, AUTOMATA 2023, Trieste, Italy, August 30 – September 1, 2023, Proceedings (pp.105-119). Springer Science and Business Media Deutschland GmbH [10.1007/978-3-031-42250-8_8].
On the Minimum Distance of Subspace Codes Generated by Linear Cellular Automata
Mariot, Luca
;
2023
Abstract
Motivated by applications to noncoherent network coding, we study subspace codes defined by sets of linear cellular automata (CA). As a first remark, we show that a family of linear CA where the local rules have the same diameter—and thus the associated polynomials have the same degree—induces a Grassmannian code. Then, we prove that the minimum distance of such a code is determined by the maximum degree occurring among the pairwise greatest common divisors (GCD) of the polynomials in the family. Finally, we consider the setting where all such polynomials have the same GCD, and determine the cardinality of the corresponding Grassmannian code. As a particular case, we show that if all polynomials in the family are pairwise coprime, the resulting Grassmannian code has the highest minimum distance possible.File | Dimensione | Formato | |
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