On the Structure of Higher Order MDS Codes
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A code of length n is said to be (combinatorially) (ρ,L)-list decodable if the Hamming ball of radius ρ n around any vector in the ambient space does not contain more than L codewords. We study a recently introduced class of higher order MDS codes, which are closely related (via duality) to codes that achieve a generalized Singleton bound for list decodability. For some ℓ≥ 1, higher order MDS codes of length n, dimension k, and order ℓ are denoted as (n,k)-MDS(ℓ) codes. We present a number of results on the structure of these codes, identifying the `extend-ability' of their parameters in various scenarios. Specifically, for some parameter regimes, we identify conditions under which (n_1,k_1)-MDS(ℓ_1) codes can be obtained from (n_2,k_2)-MDS(ℓ_2) codes, via various techniques. We believe that these results will aid in efficient constructions of higher order MDS codes. We also obtain a new field size upper bound for the existence of such codes, which arguably improves over the best known existing bound, in some parameter regimes.
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