Hadamard codes which have an even number of ones in each half

A.E. Brouwer, E.C. Posner, Z. Reichstein

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

This paper considers a particular case of the problem of binary codes with the constraint that the codes restricted to cerl3in subsets of columns must be contained in particular codes of the shorter lengths. Here we consider codes of even length 2 k, and of minimum distance ~ d. However, the code obtained by restricting to the first k positions must have even weight and at the same time the code obtained by restricting to the last k positions also must have even weight. If k = 2 n, so that the length is 4 n, n odd, and d = 2 n, we prove that the code has at most 8 n - 4 codewords. Further, 8n - 4 is attainable if and only if a Hadamard matrix of size 4 n exists. For n = 3, this yields 20 binary words of length 12 and distance ~ 6. where the number of 1 s in the first six and in the last six positions is even for every codeword in the code. This permits a file·transfer protocol control ftmction assignment for personal computers to be chosen for 20 control functions using what amounts to" pairs of upper-case alphabetic ASCII characters. In this case, the Hamming distance between the binary fomlS of every tWO different control fwtctions is at least six.
Original languageEnglish
Pages (from-to)131-140
JournalUtilitas Mathematica
Volume32
Publication statusPublished - 1987

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