On the maximal length of near-MDS codes
DOI:
https://doi.org/10.55630/mem.2026.55.405-413Keywords:
near-MDS codes, MDS codes, elliptic curves, (n, 3)-arcs, n-tracks, projective geometries, designsAbstract
This paper summarizes the known results about near-MDS codes proved in the past thirty years. The main focus is put on the problem of determining the value of the function m'(k, q) defined as the maximal length of a near-MDS code of dimension k over the field with q elements. For dimensions k > q +2 we improve the upper bound m'(k, q) <= 2q+k−2 which follows from the nonexistence of maximal arcs over fields of odd characteristic. In analogy with the main conjecture for MDS codes we formulate some conjectures on the exact value of m'(k, q).
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