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Article Dans Une Revue SIAM Journal on Discrete Mathematics Année : 2022

Three Combinatorial Perspectives on Minimal Codes

Résumé

We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-Füredi Theorem and the Combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. Applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.
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hal-03852308 , version 1 (14-11-2022)

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Gianira Alfarano, Martino Borello, Alessandro Neri, Alberto Ravagnani. Three Combinatorial Perspectives on Minimal Codes. SIAM Journal on Discrete Mathematics, 2022, 36 (1), pp.461-489. ⟨10.1137/21M1391493⟩. ⟨hal-03852308⟩
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