The set of permutations generated by cyclic shift is studied using a number system coding for these permutations. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code describing structural and symmetry properties of the set of permutations ordered according to generation by cyclic shift. The code is...

Abstract Using algebraic techniques, one of the co-authors has designed a [155,64,20] low density parity check code based on permutation matrices. This code is quasi-cyclic by construction and hence admits a convolutional representation. A set of low density parity check convolutional codes is derived from this quasi-cyclic code and its generalizations. We investigate the performance of these c...

Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study -QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). ...

Let r, s, t be three positive integers and C be a binary linear code of lenght r + s + t. We say that C is a triple cyclic code of lenght (r, s, t) over Z 2 if the set of coordinates can be partitioned into three parts that any cyclic shift of the coordinates of the parts leaves invariant the code. These codes can be considered as Z 2 [x]-submodules of Z2[x] x r −1 × Z2[x] x s −1 × Z2[x] x t −1...

Frequency-domain equalization (FDE) can take advantage of the frequency-selectivity of the channel to improve the transmission performance in a frequency selective fading channel. To further improve the transmission performance, the transmit diversity technique can be used. Cyclic delay transmit diversity (CDTD) can strengthen the frequencyselectivity while space-time transmit diversity (STTD) ...

Almwct--Two new computationally efficient afgorithms are developed for fmdiug tke exact burst-comxtiug limit of a cyclic code. The first algorithm is based on testing tbe culumn rank of certain submatrices of the parity& matrix of the code. Aa auxiliary result is a proof that every cyclic (n,k) code, with a min immu distauce of at least three, CorrecQ at least all bursts of length L(n-2k+l)/2J ...

We consider the multi-source network coding problem in cyclic networks. This problem involves several difficulties not found in acyclic networks, due to additional causality requirements. This paper highlights the difficulty of these causality conditions by analyzing two example cyclic networks which are structurally similar. Both networks have an essentially identical network code which appear...

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three zeros. Recently, [37] constructed a class of cyclic codes with arbitrary number of zeros, and computed the weight distributions for several cases. In this paper, we determine the weight distribution ...

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length lm and ideals of Ml(Fq)[X]/(X m −1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study th...

Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix G has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, usi...