On Instantaneous Codes

Maximal instantaneous codes are characterized by the property that they allow unique parsing of every infinite string. The sequence of codeword lengths of a maximal instantaneous code, sequenced in lexicographic order of the codewords, completely determines the code itself. Any increasing, decreasing or unimodal reordering of such a sequence again corresponds to a maximal instantaneous code. Lexicographic length sequences are characterized by a family of Kraft-type equalities. 1. Strings and parsing For general definitions and background we refer to Roman [R]. For greater precision and to accommodate some generalizations we make explicit a few definitions as we understand them. If A is any set, called the alphabet, then a string or word over A is a map I → A where I is a set of positive integers such that i ∈ I, 1 ≤ j ≤ i implies j ∈ I. If I is finite then the number | I | of its elements is called the length of the string. For I = ∅ we have the empty string, denoted θ. For infinite I, I = {1, 2, 3, ...}, we speak of an infinite string. A string a is a prefix of a string b if a as a map is a restriction of b. A string a : I → A is also denoted by (ai : i ∈ I), where ai = a(i), or by (a1, a2, a3, . . .) or a1a2a3 . . . A code over A is a set C of finite words over A. Members of C are called C-words, or codewords when C is understood from the context. A code C is instantaneous if no codeword is a prefix of another, distinct codeword. A maximal instantaneous code (over a given alphabet) is an instantaneous code that is not contained properly in any larger instantaneous code (over the same alphabet). If (ai)i∈I is a string of finite length strings, ai : Ii → A, then the concatenation a = a1a2 . . . is defined to be the string a : J → A given by