We show that every [n, k, d]3 code with diversity (Φ0, Φ1), 3 ≤ k ≤ 5, gcd(d, 3) = 1, is (2, 1)-extendable except for the case (Φ0, Φ1) = (40, 36) for k = 5, and that an [n, 5, d]3 code with diversity (40, 36), gcd(d, 3) = 1, is (2, 1)-extendable if Ad ≤ 50. Geometric conditions for the (2, 1)-extendability of not necessarily extendable [n, k, d]3 codes for k = 5, 6 are also given.

In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), highest common factor (hcf), or greatest commonmeasure (gcm), of two or more integers (when at least one of them is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2] This notion can be extended to polynomials, see ...

Let Fq be a finite field with q = p , where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2p over Fq, a ≥ 1. The construction of self-dual negacyclic codes of length 2bp over Fq is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.

Based on the Bezout approach we propose a simple algorithm to determine gcd of two polynomials that don't need division, like Euclidean algorithm, or determinant calculations, Sylvester matrix algorithm. The needs only n steps for degree n. Formal manipulations give discriminant resultant any without needing division calculation.

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. We present an algorithm based on a version of structured nonlinear total least squares (SNTLS) meth...