Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]
Abstract: We investigate the functional code introduced by G. Lachaud (1996) in the special case where X is a non-singular Hermitian variety in and . In , F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) on the minimum distance of this code for a Hermitian variety X in . In this paper, we will answer the question about the minimum distance in general dimension N, with . We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes. [Copyright &y& Elsevier]
Abstract: In this paper we treat cyclotomic binary duadic codes. The conjecture of Ding and Pless is that there are infinitely many cyclotomic duadic codes of prime lengths that are not quadratic residue codes. We shall prove this conjecture by using the special case of Tschebotareff''s density theorem. [Copyright &y& Elsevier]
Let p be a prime, u be a linear recurring sequence of integers of order d and let S=3d2+9d/2+1. The main result of the present paper: if u is uniformly distributed mod pS, then it is uniformly distributed mod ps for all s⩾1. This solves a longstanding folklore conjecture. [Copyright &y& Elsevier]