There is a one-to-one correspondence between l-quasi-cyclic codes over a finite field Fq and linear codes over a ring R = Fq[Y ]/(Y m − 1). Using this correspondence, we prove that every l-quasi-cyclic self-dual code of length ml over a finite field Fq can be obtained by the building-up construction, provided that char (Fq) = 2 or q ≡ 1 (mod 4), m is a prime p, and q is a primitive element of F...

Given a set H of binary vectors of length n, is there a cyclic listing of H so that every two successive vectors differ in a single coordinate? The problem of existence of such a listing, which is called a cyclic Gray code of H, is known to be NP-complete in general. The goal of this paper is therefore to specify boundaries between its intractability and polynomial decidability. For that purpos...

In this paper, we give a natural way of deciding whether a given cyclic code contains a word of given weight. The method is based on the manipulation of the locators and of the locator polynomial of a codeword x. Because of the dimensions of the problem, we need to use a symbolic computation software, like Maple or Scratchpad II. The method can be ineeective when the length is too large. The pa...

We consider a class of 3-error-correcting cyclic codes of length 2−1 over the two-element field F2. The generator polynomial of a code of this class has zeroes α, α i +1 and α j , where α is a primitive element of the field F2m . In short, {1, 2+1, 2 +1} refers to the zero set of these codes. Kasami in 1971 and Bracken and Helleseth in 2009, showed that cyclic codes with zeroes {1, 2 + 1, 2 + 1...

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let m = 2l+1 for an integer l≥ 1 and π be a generator of GF(3m)∗. In this paper, a class of cyclic codes C(u,v) over GF(3) with two nonzeros πu and πv is studied, where u = (3m +1)/2, and v = 2 ·3l+ 1 ...

The key objective of this paper is to study the cyclic codes over mixed alphabets on structure FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings αi in Fq/{0} for i∈{1,2}, q=pm with m≥1 a positive integer p an odd prime. Moreover,with applications, we obtain better new quantum error-correcting (QEC) codes. For another application ring P, several optimal help Gr...

In this paper we present a method to search q circulant matrices; the concatenation of these circulant matrices with circulant identity matrix generates quasi-cyclic codes with high various code rate q/(q+1) (q an integer). This method searches circulant matrices in order to find the good quasi-cyclic code (QCC) having the largest minimum distance. A modified simulated annealing algorithm is us...

3 Quantum Error Correction 13 3.1 Trivial Two-Qubit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Quantum Repetition Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Quantum Code for a Cyclic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Three Quantum Spin-Half Particles . . . . . . . . . . . . . . . ....

This survey paper provides fundamentals in the design of LDPC codes. To provide a target for the code designer, we first summarize the EXIT chart technique for determining (near-)optimal degree distributions for LDPC code ensembles. We also demonstrate the simplicity of representing codes by protographs and how this naturally leads to quasi-cyclic LDPC codes. The EXIT chart technique is then ex...