Theoretical construction and algorithms specifying determinants of ±1 matrices of the spectrum.

In this paper, algorithms are presented constructing, for the Hadamard maximum determinant problem of dimension n with elements ± 1. This leads to the construction of efficient algorithms specifying the determinants of n × n matrices with elements ± 1. The notion of lexicographically ordered sequences of integers and matrices is presented and we describe algorithms creating all possible lexicographically ordered vectors of dimension. With the ordered vectors, we create all the alternative determinants and search for their values, so the ones we meet first are the lexically smaller ones. The implementation of the algorithms is introduced, both Exhaustive Search, Brute force Algorithm and Partial Search and we compare these algorithms according to their speed and efficiency. Furthermore, we present some theoretical constructions attaining the evaluation of the spectrum of matrices with elements ± 1. Summarizing in this article, we develop theoretical techniques on which we can rely and build faster algorithms to find the determinants of ± 1 ± 1. matrices of the spectrum. [ABSTRACT FROM AUTHOR]