The Structure and Properties of Binary Cyclic Alphabets

In this paper the word alphabet denotes a systematic code -- one in which each code word contains a certain fixed number, k, of information places, the contents of which are arbitrary, and a fixed number, n -- k, of parity check places. Each parity check digit is the sum of the contents of a particular subset of the information places. The number n is called the block length of the alphabet. The individual members of the alphabet are called letters. 303 304 T H E BELL SYSTEM TECHNICAL J O U R N A L , F E B R U A R Y 19(55 It is well known1 that the letters of an alphabet of block length n form a subspace of the vector space of all possible rows of n binary symbols. This large space is denoted by V", V being the field 0, 1. The number of information places, 1c, is also the dimension of the subspace occupied by the alphabet. 1 A cyclic alphabet has the additional property that, if it contains a letter a, it contains as well every vector of V" which is a cyclic permutation of a. Cyclic alphabets are popular for error control for several good reasons. First, it is easy and relatively inexpensive to encode a cyclic alphabet. Second, the "best" known alphabets are cyclic alphabets.* Third, the cyclic property introduces a great deal of algebraic structure, which may be used to predict the error-detecting properties of the alphabet and to find alphabets with appropriate properties. An alphabet to be used in a data transmission system must satisfy certain requirements. There will certainly be restrictions on the size of n and k, and one naturally requires also that the alphabet should be of some use for error control.