Showing papers by "Richard A. Brualdi published in 1990"
TL;DR: This work introduces some new parameters related to the exponent of primitive digraphs with n vertices and obtains bounds on these parameters.
Abstract: The exponent of a primitive digraph is the smallest integer t such that for each ordered pair of (not necessarily distinct) vertices x and y there is a path of length t from x to y. There is considerable information known about bounds on exponents and those numbers that can be exponents of primitive digraphs with n vertices. We introduce some new parameters related to the exponent and obtain bounds on these parameters.
64 citations
01 Jan 1990
34 citations
TL;DR: In this article, the existence of a family of first-order Reed-Muller codes R(1, m) with orphan cosets is proved. But the existence is not proved for m > m.
Abstract: If C is a code, an orphan is a coset that is not a descendant. Orphans arise naturally in the investigation of the covering radius. Case C has only even-weight vectors and minimum distance of at least four. Cosets that are orphans are characterized, and then the existence is proved of a family of orphans of first-order Reed-Muller codes R(1, m). For m >
23 citations
TL;DR: It is shown how Jurkat and Ryser's matrix factorizations of the determinant and the permanent follow from an elementary proposition about graded partially ordered sets.
21 citations
01 Jan 1990
TL;DR: This work develops techniques for showing the non-existence of short codes with a given covering radius and concludes with a table which gives the best available information for the length of a code with codimension m and covering radius r.
Abstract: We further develop techniques for showing the non-existence of short codes with a given covering radius. In particular we show that there does not exist a code of codimension 11 and covering radius 2 which has length 64. We conclude with a table which gives the best available information for the length of a code with codimension m and covering radius r for 2 ≤ m ≤ 24 and 2 ≤ r ≤ 24.
14 citations
7 citations
TL;DR: An improved inequality for large i is obtained and the exact range of values for ϱ ( C 0 ) and ϰ ( C ) are determined.
7 citations
01 Jan 1990
TL;DR: In this article, it was shown that any complex square matrix T is a sum of finitely many idempotent matrices if and only if tr T is an integer and tr T > rank T.
Abstract: We show that any complex square matrix T is a sum of finitely many idempotent matrices if and only if tr T is an integer and tr T > rank T. Moreover, in this case the idempotents may be chosen such that each has rank one and has range contained in that of T. We also consider the problem of the minimum number of idempotents needed to sum to T and obtain some partial results. A complex square matrix T is idempotent if T” = T. In this paper, we characterize matrices which can be expressed as a sum of finitely many idempotent matrices and consider the minimum number of idempotents needed in such expressions. In the following, tr T denotes the trace of a matrix T, ran T denotes its range, rank T the dimension of ran T, and ker T the kernel of T. The n x n identity matrix is denoted by I,, or 1 if the size is not emphasized. Similarly for the zero matrix: 0, or 0. Two matrices T and S are similar, denoted T = S, if XT = SX for some nonsingular matrix X; they are unitarily equivalent, T z S, if the above X can be chosen to be unitary. If T and S act on spaces H and K, respectively, then