Perfect codes and balanced generalized weighing matrices. II
A balanced generalized weighing matrix \(BGW(m,k,\mu)\) over a multiplicative given group \(G\) is an \(m\times m\) matrix \(W= (w_{ij})\) with entries from \(G\cup\{0\}\) such that each row of \(W\) contains exactly \(k\) nonzero entries, and for every \(a,b\in \{1,\dots, m\}\), \(a\neq b\), the multiset \(\{w_{ai}w_{bi}^{-1}: 1\leq i\leq m,w_{ai},w_{bi}\neq 0\}\) contains exactly \(\mu/|G|\) copies of each element of \(G\). Two matrices over a field \(F\) are said to be monomially equivalent if one is obtainable from the other by permutations of rows and columns and multiplying rows and columns by nonzero elements from \(F\). The present authors are concerned with the ``classical'' series of \(BGW\)-matrices with parameters \(((q^d-1)/(q- 1),q^{d-1},q^{d-1}- q^{d-2})\) over the cyclic group of order \(q-1\), where \(q\) is a prime power and \(d\geq 2\) an integer. The authors show that there is a wealth of monomially inequivalent \(BGW\)-matrices with these parameters that are distinguishable by their rank over \(GF(q)\). A general construction yielding infinite families of matrices with various ranks is given. These examples are obtained by applying suitable automorphisms of the cyclic group \(GF(q)^*\) to the matrices constructed in Part I [ibid. 5, 294-300 (1999; Zbl 1033.94014)] by using single-error-correcting perfect codes and which were characterized as the matrices of minimum \(q\)-rank \(d\). In this paper the authors 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. [See also an electronic version with the same title, WCC2001 international workshop coding cryptography, Paris, France 2001, Electron. Notes Discrete Math. 6 (2001; Zbl 1004.94549)].
scientific article; zbMATH DE number 1781823
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