The equivalence of codes

Abstract

When discussing equivalent codes, the literature tends to define equivalence differently for linear or nonlinear codes. For non-linear codes, one normally defines equivalence based on a permutation of coordinates and permutations of the code alphabet on each coordinate. On the other hand, for linear codes, equivalence usually specifies that the permutations on each coordinate also be linear maps. The first part of this thesis proves that two linear codes are equivalent by the general definition if and only if they are equivalent by the linear definition, and also extends this statement to additive MDS codes. The second part of this paper describes a polynomial-time reduction from isomorphism of multigraphs to additive code equivalence.