Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f . Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polyn...

There is a large gap between Zhang’s theoretical bound for the length n of a binary extremal self-dual doublyeven code and what we can construct. The largest n is 136. In order to find examples for larger n a non-trivial automorphism group might be helpful. In the list of known examples extended quadratic residue codes and quadratic double circulant codes have large automorphism groups. But in ...

The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field Fq gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation b...

We consider linear codes in the metric space with the Niederreiter-Rosenbloom-Tsfasman (NRT) metric, calling them linear ordered codes. In the first part of the paper we examine a linear-algebraic perspective of linear ordered codes, focusing on the distribution of “shapes” of codevectors. We define a multivariate Tutte polynomial of the linear code and prove a duality relation for the Tutte po...

A hstract-A partial classification is given of the self-dual codes of length 24 over GF(3). The main results are as follows: there are exactly two codes with minimum Hamming distance d = 9; most of the codes have d = 6 and are indecomposable; one code with d = 6 has a trivial automor-phism group (this is the first such self-dual code that has been found); the codes generated by the 59 inequival...

In his celebrated paper on the algebraic structure of ConvoJutional codes, Forney 111 showed that by using the invariant-factor theorem, one can transform an arbitrary polynomial generator matrix for an (n, k ) convolutional code C into a basic (and ultimately a minimal) generator matrix for C. He also showed how to find a polynomial inverse for a basic generator matrix for C, and a basic gener...

Let m, d and k be positive integers such that k ≤ me , where e = (m, d). Let p be an prime number and π a primitive element of Fpm . To each ~a = (a0, · · · , ak−1) ∈ F k pm , we associate the linearized polynomial f~a(x) = k−1 ∑ j=0 ajx p . And, to each f~a(x), we associated the sequence c~a = (f~a(1), f~a(π), · · · , f~a(π pm−2)). Let C = {c~a | ~a ∈ F k pm} be the cyclic code formed by the s...

We propose a new method to construct quantum synchronizable codes from classical cyclic codes using the idea of augmented cyclic codes. The method augments a dual-containing cyclic code C2 to obtain another cyclic codes C1 of higher dimension. The resulting two cyclic codes and the dual code C⊥ 2 satisfy the containing property C⊥ 2 ⊂ C2 ⊂ C1. The proposed construction method based on general q...

Let R = Z4[v]/〈v 2 + 2v〉 = Z4 + vZ4 (v 2 = 2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z4-linear Gray map from R onto Z 2 4 which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classif...

The existence of a quantum [[28, 12, 6]] code was one of the few cases for codes of length n ≤ 30 that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result of this paper is the construction of the first optimal linear quaternary [28, 20, 6] code which contains its Hermitian dual code and yields the first optimal quantum [[28, 12, 6]] code. 2010 MSC: 94B...