Perfect Codes and Balanced Generalized Weighing Matrices, II

✍ Scribed by Dieter Jungnickel; Vladimir D. Tonchev

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 116 KB
• Volume: 8
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.2001.0327

✦ Synopsis

In a previous paper, the authors proved that any set of representatives of the distinct 1dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (qB!1)/(q!1) over GF(q) is a balanced generalized weighing matrix over the multiplicative group of GF(q). Moreover, this matrix was characterized as the unique (up to monomial equivalence) weighing matrix for the given parameters with minimum q-rank. We now relate these matrices to m-sequences (that is, linear shift register sequences of maximal period) by giving an explicit description in terms of the trace function; in this way, we show that a simple modi"cation of our method can be used to obtain the matrices which are given by the &&classical,'' more involved construction going back to Berman. Moreover, further modi"cations of our matrices actually yield a wealth of monomially inequivalent examples, namely matrices for many di!erent q-ranks.