Abstract
                                                                        A code of length n is said to be (combinatorially)  (ρ, L)-list decodable if the Hamming ball of radius ρn 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 (n1, k1)-MDS(ℓ1) codes can be obtained from  (n2, k2)-MDS(ℓ2) 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.  Due to space restrictions, the full version of this paper  containing proofs of claims is made available in [1].