For simplicity, we adopt the following convention: i, j, k, n, l denote elements of N, K denotes a field, a, b, c denote elements of K, p, q denote finite sequences of elements of K, and M1, M2, M3 denote square matrices over K of dimension n. Next we state two propositions: (1) 1K · p = p. (2) (−1K) · p = −p. Let K be a set, let M be a matrix over K, and let p be a finite sequence. We say that...

A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74–88. © 2006 Elsevier B.V. All rights reserved.

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order n— i.e., to characterize entire functions f with the property that f(A) is entrywise nonnegative for every entrywise nonnegative matrix A of size n×n. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) uppertriangular matri...

In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW (31, 16), one matrix in CW (21, 16) and another one in CW (63, 16) (not obtainable by Kronecker product from CW (21, 16)). The classification uses a multiplier existence t...

We disprove the following conjecture due to Vı́ctor Neumann-Lara: for every pair (r, s) of integers such that r > s > 2, there is an infinite set of circulant tournaments T such that the dichromatic number and the cyclic triangle free disconnection of T are equal to r and s, respectively. Let Fr,s denote the set of circulant tournaments T with dc(T ) = r and − →ω 3 (T ) = s. We show that for eve...

The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.

Mobeen Munir 1, Waqas Nazeer 1, Zakia Shahzadi 1 and Shin Min Kang 2,3,* 1 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] (M.M.); [email protected] (W.N.); [email protected] (Z.S.) 2 Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea 3 Center for General Educati...

A critical set in a latin square is a subset of its elements with the following properties: 1) No other latin square exists which also contains that subset. 2) No element may be deleted without destroying property 1. Let scs(n) denote the smallest possible cardinality of a critical set in an n × n latin square. It is conjectured that scs(n) = n/4 , and that only the back-circulant latin square ...