Search

Let G = (V , E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0 s 1 1 t 1 ... 0 s k 1 t k , where s i and t i are positive integers with 1 ≤ i ≤ k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled "Which graphs have integral spectra?" in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0 s 1 1 t 1 ... 0 s k 1 t k . We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p , we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Z p m (where m is a positive integer) and the polynomial ring Z p [ x ] / (x p). [ABSTRACT FROM AUTHOR]

This paper constructs a class of lattices whose discrete analogues are variable-length non-linear codes. The well-known discrete analogue of lattices and linear codes inspires our approach. We next design a variable length binary non-linear code called automorphism orbit code from a finite abelian p -group of rank greater than 1, where p is a prime. For each finite abelian p -group, codewords of the automorphism orbit code are variable-length codewords called automorphism orbit codewords. The homomorphisms between groups determine homomorphism codes , whereas automorphism orbit codes are specified by partitions of a number, orbits of group action, homomorphisms and automorphisms of groups. For some groups G and H , we shall use elements of H o m (G , H) to create a cover relation for bit strings of codewords of an automorphism orbit code and formulate a lattice of variable length non-linear codes. [ABSTRACT FROM AUTHOR]

In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming an upper bound on the order of the monomial automorphism groups, according to computational results. [ABSTRACT FROM AUTHOR]

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z 4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z 4 -linear codes and good binary nonlinear codes are constructed from GQC codes over Z 4. [ABSTRACT FROM AUTHOR]

In this paper, we always assume that p and n > 3 are two distinct odd primes, m ≥ 2 is a positive integer, and gcd ⁡ (p m , 2 n (2 2 n − 1)) ≠ 1. Set r = p 2 or r = p 3 , we determine upper bounds on the number of inequivalent extended binary irreducible Goppa codes Γ (L , g) of degree r and length 2 n + 1 , where L = F 2 n ∪ { ∞ } is the support set and each g (x) is an irreducible polynomial of degree r over F 2 n . [ABSTRACT FROM AUTHOR]