Cyclic Product Codes Research Articles

New bounds on the minimum distance of cyclic codes

Two bounds on the minimum distance of cyclic codes are proposed. The first one generalizes the Roos bound by embedding the given cyclic code into a cyclic product code. The second bound also uses a second cyclic code, namely the non-zero-locator code, but is not directly related to cyclic product codes and it generalizes a special case of the Roos bound.

Spectral Analysis of Quasi-Cyclic Product Codes

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row and column codes. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm. More specifically, we prove an explicit (unreduced) basis of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code in terms of the generator matrix in reduced Grobner basis with respect to the position-over-term (RGB/POT) order form of the $ {\ell _{A}}$ -quasi-cyclic row code and the $ {\ell _{B}}$ -quasi-cyclic column code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where $ {\ell _{A}}= {\ell _{B}}=1$ ). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code for two special cases: i) for $ {\ell _{A}}=2$ and $ {\ell _{B}}=1$ and ii) if the row code is a one-level $ {\ell _{A}}$ -quasi-cyclic code (for arbitrary $ {\ell _{A}}$ ) and $ {\ell _{B}}=1$ . For arbitrary $ {\ell _{A}}$ and $ {\ell _{B}}$ , the Pre-RGB/POT form of the generator matrix of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code is conjectured. The spectral analysis is applied to the generator matrix of the product of an $ {\ell }$ -quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given $ {\ell }$ -quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for $ {\ell }$ -phased burst errors with guaranteed decoding radius.

Decoding of repeated-root cyclic codes up to new bounds on their minimum distance

The well-known approach of Bose, Ray-Chaudhuri, and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum Hamming distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined.

Doubly-periodic sequences and a class of two-dimensional cyclic codes

In this paper we continue our study of doubly-periodic sequences. It is shown that the set of eventually doubly-periodic sequences over a finite field has the structure of a Hopf algebra and that it factors as the tensor product of the Hopf algebra of eventually singly periodic sequences with itself. As an application of this result, we generalize a previous theorem by introducing a coalgebra structure for a space of doubly-periodic sequences (which solve a two-dimensional recurrence) and decomposing it as a sum of products of coalgebras of singly periodic sequences. The result is further applied to the well-known construction of two-dimensional cyclic product codes using cyclic ordering yielding an induced coalgebra structure for these codes.

Multiple-burst error-correcting cyclic product codes (Corresp.)

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> be the cyclic product code of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> single parity check codes of relatively prime lengths <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{1}, n_{2},\cdots , n_{p} (n_{1} &lt; n_{2} &lt; \cdots &lt; n_{p})</tex> . It is proven that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> can correct <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{P-2}+2^{p-3}-1</tex> bursts of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{1}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lfloor(\max\{p+1, \min\{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}\}\}-1)/2\rfloor</tex> bursts of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{1}n_{2} \cdots n_{s} (2\leq s \leq p-2)</tex> . For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p=3</tex> this means that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> is double-burst- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{1}</tex> -correcting. An efficient decoding algorithm is presented for this code.

A powerful concatenated matrix code using cyclic product and matrix codes

In this paper a new class of block concatenated matrix codes is investigated. This concatenated code in closely related in structure to a matrix type-B code constructed with a cyclic product inner code and a matrix code of type-A or iterative code as an outer code. This structure is called a cyclic product concatenated matrix type-B code. This presents a possible method that has become feasible for handling incomplete (erroneous) data that arises in noisy data communication links.

Optimal encodings of linear block codes for unequal error protection

It is possible for a linear block code to provide more protection for selected message positions than is guaranteed by the minimum distance of the code. The protection provided a message position can be measured by associating a number with that position called its separation. The separation of a message position measures the protection provided to that position in a manner analogous to that in which the minimum distance of a code measures the protection provided the entire message. This paper proves that any fixed linear block code has an encoding which is optimal with respect to the error protection provided the individual message positions. More precisely, among those encodings of the code for which the separations associated with the message positions are arranged in nondecreasing order, there is at least one which simultaneously maximizes all the separations associated with the message positions. A procedure is given which may be used to construct optimal encodings for linear codes of small dimension. When the Hamming metric is employed, the procedure builds a generator matrix which is as sparse as possible for the given code. At each iteration the procedure adds a row to a partially constructed generator matrix. A code word of minimum weight is chosen for this purpose—subject to the restriction that the rows of the generator matrix must be linearly independent. A more general result is that any generator matrix which is as sparse as possible induces an optimal encoding of its row space. A similar result holds when the Lee metric is used to model a channel. Theorems dealing with cyclic codes and product codes are developed. Under suitable restrictions, an optimal generator matrix for a cyclic code may be formed by concatenating the generator matrices of the minimal ideals which are contained in it. When the Hamming metric is employed, an optimal generator matrix for a product code may be obtained by taking the Kronecker product of optimal generator matrices for the component codes.

Single- and multiple-burst-correcting properties of a class of cyclic product codes

The direct product of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> single parity-check codes of block lengths <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1,n_2, \cdots ,n_p</tex> is a cyclic code of block length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1 \times n_2 \times \cdots \times n_p</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n_1 - 1) \times (n_2 - 1) \times \cdots \times (n_p - 1)</tex> information symbols per block, if the integers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1,n_2 \cdots ,n_p</tex> are relatively prime in pairs. A lower bound for the single-burst-correction (SBC) capability of these codes is obtained. Then, a detailed analysis is made for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = 3</tex> , and it is shown that the codes can correct one long burst or two short bursts of errors. A lower bound for the double-burst-correction (DBC) capability is derived, and a simple decoding algorithm is obtained. The generalization to correcting an arbitrary number of bursts is discussed.

Further results on cyclic product codes

Cyclic product codes are useful for two reasons. First, they impart a great deal of algebraic structure to a subclass of the class of cyclic codes. Second, because they can be formulated in terms of much shorter (component) codes, their decoding may be considerably simpler than many other types of codes. In this paper both of the properties of cyclic product codes are developed. It is shown that the product of two majority-logic decodable cyclic codes is also majority-logic decodable provided that one of the component codes is one-step decodable. More precisely, if the row-component code can realize minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_1</tex> (i.e., correct <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[(d_1 -- 1)/2]</tex> errors) with a one-step majority-logic decoder and if the column-component code can realize minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_2</tex> with an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -step decoder, then the product code can realize distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_1 d_2</tex> with an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -step decoder. It is also shown that the algebraic structure of cyclic product codes can be applied to establish the exact minimum distance of certain subclasses of BCH codes.

Cascade Decoding of Cyclic Product Codes

In this paper, it is shown how to synthesize simple decoders for cyclic product codes. Cyclic product codes may be synthesized in the form of interlaced codes if the block lengths of the codes are relatively prime. In this case, the decoder can be synthesized in the form of a cascade of decoders, each of which operates on one of the subcodes forming the product code. The cascade decoders described differ from most of the decoders given in the coding literature in two major respects. First, they can be built. Second, instead of correcting all error patterns with weight less than some fixed value and no error patterns of greater weight, cascade decoders correct many error patterns beyond their guaranteed correction capability. Thus, the effective error-correction performance using a cascade decoder may be considerably beyond the usual Elias bound for product codes.

Cyclic product codes and their implementation

In many communication systems, variable-redundancy coding schemes are highly desirable either due to the fact that messages of various degrees of importance are present, or due to some change in the real communication channel. A class of “cyclic product codes” presented in this paper is capable of operating in variable redundancy modes. In the simplest case, the generator polynomial in the high redundancy mode is g(x) = g 1 (x)g 2 (x) , while the generator polynomial in the low redundancy mode is only g 1 (x) . It is shown that efficient product codes can be constructed offering different degrees of protection against independent errors, burst errors, and multiple burst errors. It is also shown that particularly simple implementation for cyclic product codes is possible. In fact, the complexity of the entire decoder can be made roughly the same as that of the decoder for the high redundancy code alone. Hence, the implementation of low-redundancy codes is accomplished with little extra cost.

Cyclic product codes

A new class of cyclic codes, cyclic product codes, is characterized. These codes enjoy the implementation advantages of cyclic codes and, in addition, possess the important structural properties of product (iterated) codes. The main results are as follows: \begin{enumerate} \item Conditions are given which ensure that the product of two, and, hence, arbitrarily many, cyclic codes is itself a cyclic code. \item Cyclic product codes are shown to be capable of unambiguous correction of both bursts and random errors. \item The generator polynomial of the cyclic product code is derived and shown to be a simple function of the generator polynomials of the subcodes. \end{enumerate} These results have the following various applications: a) the codes effect a compromise between random and burst-error-correcting codes and, therefore, appear to be well-suited to error correction on channels in which both types of errors occur, b) many codes in a particular subclass of the class of cyclic product codes are efficient burst-error correctors and are more easily implemented than the equivalent codes formed by interleaving short codes, c) the previously mentioned results 1) and 3) can be applied to some cyclic codes to show that they are product codes as well; then known results on the minimum distance of product codes can be applied. This improves on the Bose-Chaudhuri-Hocquenghem lower bound in many cases.