Relativized codes

A code C over an alphabet Σ is a set of words such that every word in C + has a unique factorization over C, that is, a unique C-decoding. When not all words in C + appear as messages, a weaker notion of unique factorization can be used. Thus we consider codes C relative to a given set of messages L, such that each word in L has a unique C-decoding. We extend this idea of relativizing code concepts to restricted message spaces. In general terms, from a predicate P defining a class of codes, P-codes, we derive a relativized version of such codes, P-codes relative to a given language L. In essence, C⊆Σ + is a P-code relative to L⊆Σ + if P is true on its domain restricted to L. This systematic approach leads to the relativization of the definitions of many classes of codes, including prefix, suffix, bifix and solid codes. It can also be applied to certain classes of languages, like overlap-free languages, which are not codes, but which can be defined using a similar logical framework. In this paper, we explore the mechanism of this relativization and compare it to other existing methods for relativizing code properties to restricted message spaces. © 2012 Elsevier B.V. All rights reserved.