Nonbinary BCH Codes Research Articles

Two families of BCH codes and new quantum codes

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths $$n=\frac{q^{2m}-1}{q^2-1}$$ and $$n=\frac{q^{2m}-1}{q+1}$$ ( $$m\ge 3)$$ over the finite field $$\mathbf {F}_{q^2}$$ are studied. The maximum designed distances $$\delta ^\mathrm{new}_\mathrm{max}$$ of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.

Improved Constructions for Nonbinary Quantum BCH Codes

In this work, we present two improved constructions for nonbinary quantum BCH codes of lengths \(n=\frac {q^{4}-1}{2}\) and \(n=\frac {q^{4}-1}{q-1}\), where q is an odd prime power. Moreover, the constructed quantum BCH codes have parameters better than those obtained from other known constructions.

Hermitian dual containing BCH codes and Construction of new quantum codes

Let $q\geq 3$ be a prime power. Maximal designed distances of primitive Hermitian dual containing $q^{2}$-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.

Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes

Two code constructions generating new families of good nonbinary asymmetric quantum BCH codes and good nonbinary subsystem BCH codes are presented in this paper. The first one is derived from q-ary Steane’s enlargement of CSS codes applied to nonnarrow-sense BCH codes. The second one is derived from the method of defining sets of classical cyclic codes. The asymmetric quantum BCH codes and subsystem BCH codes here have better parameters than the ones available in the literature.

Designing fault-secure parallel encoders for systematic linear error correcting codes

We consider the open problem of designing fault-secure parallel encoders for various systematic linear ECC. The main idea relies on generating not only the check bits for error correction but also, separately and in parallel, the check bits for error detection. Then, the latter are compared against error detecting check bits which are regenerated from the error correcting check bits. The detailed design is presented for encoders for CRC codes. The complexity evaluation of FPGA implementations of encoders with various degrees of parallelism shows that their fault-secure versions compare favorably against their unprotected counterparts both with respect to complexity and the maximal frequency of operation. Future research will include the design of FS decoders for CRC codes as well as the generalization of the presented ideas to design of FS encoders and decoders for other systematic linear ECC like nonbinary BCH codes and Reed-Solomon codes.

Analysis of MFSK frequency-hopped spread-spectrum multiple-access over a Rayleigh fading channel

An explicit expression for the symbol error probability for a system using frequency-hopped spread-spectrum multiple-access with M-ary frequency shift keying (M-ary FSK) is given when the channel is undergoing Rayleigh fading. Using this expression, the performance of binary BCH codes, nonbinary BCH codes and RS codes are compared, and the best choices of code and modulation format are determined for different channel conditions. Depending on the required codeword error probability, different choices of code and modulation format are found to give the best result. Some attention is also paid to the sometimes used assumption of a noiseless channel, which in many cases is found to give inaccurate results.

High‐speed decoding of BCH codes using a new error‐evaluation algorithm

AbstractA new error‐evaluation method for computing error values in decoding Reed‐Solomon codes or nonbinary BCH codes using the Berle‐kamp‐Massey algorithm is proposed.Since the error‐evaluator polynomial used in the new method is obtained as by‐products of the computation for the error‐locator polynomial, the new method is more efficient than Forney's method whose error‐evaluator polynomial cannot be obtained as the by‐products of the error‐locator polynomial.

Nonbinary random error-correcting codes (Corresp.)

Primitive BCH codes with symbols from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q)</tex> and designed distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> have parameter values \begin{align} \text{block length} &= n = q^m - 1 \\ \text{check symbols/block} &= r \leq m(d - 1) \end{align} where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is any positive integer. For many nonbinary BCH codes (called maximally redundant codes), the maximum number of check symbols per block is required, i.e. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r = m(d - 1)</tex> . Conditions whereby a primitive nonbinary BCH code is maximally redundant are discussed. It is shown that a class of codes exists, with symbols from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q)</tex> , based upon doubly lengthened Reed-Solomon codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q^m)</tex> , having parameter values \begin{align} \text{block length} &= n = m(q^m + 1) \\ \text{check symbols/block} &= r = m(d - 1) \\ \text{designed distance} &= d \end{align} where again <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is any positive integer. Thus this class of codes extends the block length of maximally redundant codes by a multiplicative factor exceeding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , while retaining the same designed distance and same number of check symbols.

Adding Two Information Symbols to Certain Nonbinary BCH Codes and Some Applications

This paper is a compendium of results based on a simple observation: two information symbols can be appended to certain nonbinary BCH codes without affecting the guaranteed minimum distance of these codes. We give two formulations which achieve this result; the second yields information regarding the weights of coset leaders for the original BCH codes. Single-error-correcting Reed-Solomon codes with the added information symbols yield perfect codes for the Hamming metric. We use these lengthened Reed-Solomon codes as building blocks for perfect single-error-correcting codes in another metric.