Class Of BCH Codes Research Articles

The Hermitian Dual Codes of Several Classes of BCH Codes

As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> be a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -ary BCH code of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> with respect to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th primitive root of unity β over an extension field of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i>²</sub> , and let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥<i>H</i></sup> denote its Hermitian dual code, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is a prime power. If both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥<i>H</i></sup> are a BCH code with respect to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th primitive root of unity β, then <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> is called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Hermitian dually-BCH code</i> . The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.

Three classes of BCH codes and their duals

BCH codes are an important class of cyclic codes, and have wide applications in communication and storage systems. However, it is difficult to determine the parameters of BCH codes and only a few cases are known. In this paper, we mainly study three classes of BCH codes with n=qm−1,q2s−1q+1,qm−1q−1. On one hand, we provide the parameters of C(n,q,δ,1) and its dual code in some cases. On the other hand, we provide the sufficient and necessary conditions to guarantee that C(n,q,δ,2) is a dually-BCH code.

The Dual Codes of Several Classes of BCH Codes

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> is always relative to an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-th primitive root of unity <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> in an extension field of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>, and is called a dually-BCH code if its dual is also a BCH code relative to the same <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel&#x2019;nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

Efficient and Explicit Balanced Primer Codes

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing.

Some new results on dimension and Bose distance for various classes of BCH codes

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over Fq with designed distance δ=aqm−1−1(resp. δ=aqm−1q−1) for all 1≤a≤q−1, where q is a prime power and m>1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range [aqm−1q−1,aqm−1q−1+T] for 0≤a≤q−2, where T=qm+12−1 if m is odd, and T=2qm2−1 if m is even.

Narrow-Sense BCH Codes Over $ {\mathrm {GF}}(q)$ With Length $n=\frac {q^{m}-1}{q-1}$

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over $ {\mathrm {GF}}(q)$ with length $n=(q^{m}-1)/(q-1)$ . Little is known about this class of BCH codes when $q>2$ . The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length $n=(3^{m}-1)/2$ are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

A class of primitive BCH codes and their weight distribution

BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. Recently, several classes of BCH codes with length $$n=q^m-1$$n=qm-1 and designed distances $$\delta =(q-1)q^{m-1}-1-q^{\lfloor (m-1)/2\rfloor }$$ź=(q-1)qm-1-1-qź(m-1)/2ź and $$\delta =(q-1)q^{m-1}-1-q^{\lfloor (m+1)/2\rfloor }$$ź=(q-1)qm-1-1-qź(m+1)/2ź were widely studied, where $$m\ge 4$$mź4 is an integer. In this paper, we consider the case $$m=3$$m=3. The weight distribution of a class of primitive BCH codes with designed distance $$q^3-q^2-q-2$$q3-q2-q-2 is determined, which solves an open problem put forward in Ding et al. (Finite Fields Appl 45:237---263, 2017).

The dimension and minimum distance of two classes of primitive BCH codes

Cyclic Reed–Solomon codes, a type of BCH codes, are widely used in consumer electronics, communication systems, and data storage devices. This fact demonstrates the importance of BCH codes – a family of cyclic codes – in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with designed distances δ=(q−1)qm−1−1−q⌊(m−1)/2⌋ and δ=(q−1)qm−1−1−q⌊(m+1)/2⌋. The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known.

An Improved Algorithm of RS Codes Blind Recognition

The acquisition of information about channel coding sequences is the research emphasis and difficulties at present, and recognizing the channel coding parameters is the prerequisite for information acquisition. RS Codes are a class of linear block codes as well as a special class of BCH Codes and cyclic codes. It regards a group of symbols with a length of n as a unit to process with, and the n symbols are generated by k transmitting information symbols under some proper associated rules. RS Codes are widely used in various communication systems. In this paper, an algorithm of RS Codes blind recognition is researched. And a blind recognition model is established on MATLAB. At last, the paper presents the recognition rates of situations with different error rates.

Multirate Filter Bank Representations of RS and BCH Codes

This paper addresses the use of multirate filter banks in the context of error-correction coding. An in-depth study of these filter banks is presented, motivated by earlier results and applications based on the filter bank representation of Reed-Solomon (RS) codes, such as Soft-In Soft-Out RS-decoding or RS-OFDM. The specific structure of the filter banks (critical subsampling) is an important aspect in these applications. The goal of the paper is twofold. First, the filter bank representation of RS codes is now explained based on polynomial descriptions. This approach allows us to gain new insight in the correspondence between RS codes and filter banks. More specifically, it allows us to show that the inherent periodically time-varying character of a critically subsampled filter bank matches remarkably well with the cyclic properties of RS codes. Secondly, an extension of these techniques toward the more general class of BCH codes is presented. It is demonstrated that a BCH code can be decomposed into a sum of critically subsampled filter banks.

On the minimum distance of composite-length BCH codes

We derive a theorem which generalizes Theorem 3 in Chapter 9 of the book "The Theory of Error-Correcting Codes" by F.J. MacWilliams and N.J.A. Sloane (North-Holland, 1977). By this theorem, we are able to give several classes of BCH codes of composite length whose minimum distance does not exceed the BCH bound. Moreover, we show that this theorem can also be used to determine the true minimum distance of some other cyclic codes with composite-length.