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- W830625610 abstract "Kraft’s inequality is a classical theorem in Information Theory which establishes the existence of prefix codes for certain (admissible) length distributions. We prove the following generalisation of Kraft’s theorem: For every admissible infinite length distribution one can construct a maximal prefix codes whose codewords satisfy this length distribution. Prefix codes are widely used in data transmission or in (algorithmic) information theory (see [3, 4]). A set of nonempty words C ⊆ X∗ over an alphabet X is called a prefix code provided w ∈ C is not a prefix of v ∈ C, for every pair of distinct words w, v ∈ C. A classical theorem about the existence prefix codes is called Kraft’s inequality [2]. Theorem 1 (Kraft’s inequality). Let X be a finite alphabet, I ⊆ N and let f : I → N be a non-decreasing function such that ∑n∈I |X|− f (n) ≤ 1. Then there is a prefix code C = {vn : n ∈ I} ⊆ X∗ such that |vn| = f (n). Here |X| denotes the cardinality of the set X, and |v| denotes the length of the word v and ∑ n∈I |X|− f (n) ≤ 1 means that the length distribution f : I → N is admissible. The aim of this note is to show that a simple modification of Kraft’s construction (see e.g. [4]) is suitable for the construction of infinite maximal prefix codes C ⊆ X∗ whenever ∑v∈C |X|−|v| ≤ 1. Here a code C ⊆ X∗ is referred to as maximal prefix if C is a prefix code and for every prefix code C′ ⊇ C implies C′ = C. It is known that a maximal prefix code need not be maximal as a code (see e.g. [1, II. Example 3.1]). For finite codes C ⊆ X∗, however, a maximal prefix code satisfies∑v∈C |X|−|v| = 1 and is also maximal as a code. Theorem 2. Let f : N → N be a non-decreasing function such that ∑" @default.
- W830625610 created "2016-06-24" @default.
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- W830625610 date "2006-05-01" @default.
- W830625610 modified "2023-09-26" @default.
- W830625610 title "On Maximal Prefix Codes." @default.
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