Alternant codes

Abstract

A class of codes defined by the parity check matrix of a linear code of minimum distance at least t + 1 is examined and the number of parity check symbols is discussed. Determinants of this type are known as alternants (Muir and Metzler, 1930); codes corresponding to such a matrix are termed alternant codes. The parity check matrix is obtained by restricting some of its elements to subfields of GF(q to the m-th power). Minimum distance and redundancy bounds are established for these codes, and some interesting equivalence and invariance properties are derived.