I T ] 1 2 M ar 2 01 9 MDS codes over finite fields

The mds (maximum distance separable) conjecture claims that a nontrivial linear mds [n, k] code over the finite field GF (q) satisfies n ≤ (q + 1), except when q is even and k = 3 or k = q − 1 in which case it satisfies n ≤ (q + 2). For given field GF (q) and any given k, series of mds [q + 1, k] codes are constructed. Any [n, 3] mds or [n, n−3] mds code over GF (q) must satisfy n ≤ (q+1) for q odd and n ≤ (q+2) for q even. For even q, mds [q + 2, 3] and mds [q + 2, q − 1] codes are constructed over GF (q). The codes constructed have efficient encoding and decoding algorithms.