Error-Correcting Codes for Multiple-Level Transmission

Abstract

A q-level alphabet is defined as a row vector space over a finite field with q elements. The letters of the alphabet are the rows of the vector space, each consisting of n symbols from the ground field. The weight of a letter is the number of nonzero symbols it contains. The minimum weight of the letters of the alphabet, excluding zero, is denoted by d. A relationship is established between the alphabet and a set of points S in a finite projective space. There is a many-one correspondence between the letters of the alphabet and the hyperplanes of the space. The weight of a letter is simply related to the incidence of the set S with the corresponding hyperplane. Two sets of points in a finite projective space are called equivalent if they are related by a collineation of the space. Two alphabets are called equivalent if there exists between them, as vector spaces, a weight-preserving semi-isomorphism. It is shown that these definitions mean the same thing and reduce to the usual definition when q = 2. An inequality is established between the dimension of the alphabet and the parameters d, q, n. This gives a lower bound for n in terms of the other parameters. It is shown that this bound cannot be achieved by alphabets with repeated columns. A method is given for constructing a class of alphabets which attain this bound. It is shown that for the case q = 2 these are the only alphabets (in the sense of equivalence) for which the bound is attained.