Search

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]

Two families of asymmetric quantum codes of length n = q 2 m - 1 over F q 2 are constructed in this paper. By a detailed analysis of properties about q 2 -ary cyclotomic cosets modulo n, Hermitian dual-containing conditions for a family of primitive narrow-sense BCH codes are presented. Consequently, a series of asymmetric quantum BCH codes are constructed via the CSS-like construction and pairs of nested BCH codes. The parameters of new asymmetric quantum codes presented here are better than those available in the literatures before, and the real Z-distance are much larger than δ max + 1 . [ABSTRACT FROM AUTHOR]

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension. [ABSTRACT FROM AUTHOR]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths q m + 1 q + 1 and q m + 1 over the finite field F q , both of which are LCD codes. The dimensions of narrow-sense BCH codes of length q m + 1 q + 1 with designed distance δ = ℓ q m − 1 2 + 1 are determined, where q > 2 and 2 ≤ ℓ ≤ q − 1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length q m + 1 are developed for odd q , which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length q m + 1 being dually-BCH codes are presented, where q is odd and m ≢ 0 (mod 4). [ABSTRACT FROM AUTHOR]

BCH codes have been widely used in consumer devices, communication systems, and data storage systems. In this paper, we study the dimensions of four families of the BCH codes C (q , n , δ , 0) of length n = q m + 1 , where designed distances δ are given as follows. (1) δ = a q m + 1 q + 1 + 1 , where 1 ≤ a ≤ ⌊ q − 1 2 ⌋ and m is odd. (2) δ = a q m − 1 q − 1 +2, where 1 ≤ a ≤ ⌊ q − 1 2 ⌋. (3) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + (q − 2 a − 1) q m − 1 2 + 3 , where m is odd and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. (4) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + q m 2 + 4 , where m is even and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. The minimum distance of the first family of the BCH codes C (q , n , δ , 0) is determined. The Bose distances of the third and fourth families of the BCH codes C (q , n , δ , 0) are determined under some conditions. Many optimal linear codes are obtained from these BCH codes. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

In this paper, we investigate all coset leaders of primitive BCH codes for δ in the range 1 ≤ δ ≤ q m + 7 2 , which extends Liu and Shi's results. Besides, we also generalize Shi's results by proposing the maximum designed distance of non-narrow-sense( b = k 2 q 2 + k 1 q + k 0 ) primitive BCH codes which can contain their Euclidean dual. At the end, we calculate the dimension of the Euclidean dual containing non-narrow-sense primitive BCH codes and construct some quantum BCH codes. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m a positive integer. Suppose that a ≥ 2 is a factor of q m − 1 such that m is the multiplicative order of q modulo n : = q m − 1 a. Firstly, for m odd and m even, we present some necessary and sufficient conditions on dual-containing non-primitive BCH codes of length n with the maximum designed distances over F q , respectively. The results show that the maximum designed distances of dual-containing BCH codes in this paper are larger than those in [2, Theorem 3]. Secondly, we explicitly determine the dimensions of some dual-containing non-primitive BCH codes of length n , and finally, we construct some new quantum codes with relatively large minimum distances. [ABSTRACT FROM AUTHOR]

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length $n$ over $\mathbb {F}_{q}$ is always relative to an $n$ -th primitive root of unity $\beta $ in an extension field of $\mathbb {F}_{q}$ , and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta $. 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 $\delta $ 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’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances $\delta $. 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. [ABSTRACT FROM AUTHOR]

BCH codes are among the best practical cyclic codes widely used in consumer electronics, communication systems, and storage devices. However, not much is known about BCH codes with large minimum distance. In this paper, we consider narrow-sense BCH codes of length n = qm 1 N with designed distance delta = s q 1n proportional to n, where N divides qm 1 q 1 and 1 leq s leq q 1. We determine both their dimensions and minimum distances. In particular, when N = 1, the codes are primitive, with minimum distance d = s q 1 (qm 1) and dimension k = (q s)m. The general result on code dimensions is achieved by applying generating functions and inverse discrete Fourier transforms to an enumeration problem. [ABSTRACT FROM AUTHOR]

In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over F q 2 from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths n = q 4 - 1 a (a = 1 or a = q ± 1) , we construct some HDC matrix-product codes with lengths N = 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over F q ( 3 ≤ q ≤ 5 ) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound. [ABSTRACT FROM AUTHOR]

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length q2m-1q+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{q^{2m}-1}{q+1}$$\end{document} over the finite fields Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {F}}_q$$\end{document}. We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of C(n,q,δi)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta _i)}$$\end{document} (C(n,q,δi)⊥\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta _i)}^\perp$$\end{document}) for i=1,2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2$$\end{document} and C(n,q,δ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta )}$$\end{document} (C(n,q,δ)⊥\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta )}^\perp$$\end{document}) for 2≤δ≤q-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2\le \delta \le q-1$$\end{document}. What’s more, we obtain many (almost) optimal codes. [ABSTRACT FROM AUTHOR]

Unmanned aerial vehicles (UAVs) have attracted a lot of attention due to their enormous potentiality in civil and military applications over the past years. In order to allow accurate control action of UAV, a robust and real-time state estimation technique is required. In this paper, we propose a Kalman filter based UAV state estimation technique when the communication takes place over wireless links in an Internet of Things (IoT) network. We consider that a set of sensors observes the state of the UAV and transmits the observation to a control center (central server) over a distributed wireless IoT network. To deal with the communication impairments due to wireless communication links between the UAV's sensors and the IoT system components, e.g., IoT gateways, a Bose–Chaudhuri–Hocquenghem coded communication system is presented. Based on the received signals at the IoT gateways, a global state estimation technique is proposed. Performance of the proposed communication and estimation scheme is demonstrated through numerical results for different conditions. From the comparison with a conventional estimation scheme, it is observed that the proposed scheme significantly outperforms the conventional scheme in terms of state estimation and error performance. [ABSTRACT FROM AUTHOR]

Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

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 = q m − 1 , q 2 s − 1 q + 1 , q m − 1 q − 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. [ABSTRACT FROM AUTHOR]

Bose-Chaudhuri-Hocquenghem (BCH) codes have been intensively investigated. Even so, there is only a little known about primitive BCH codes, let alone non-primitive ones. In this paper, let q > 2 be a prime power, the dimension of a family of non-primitive BCH codes of length n = q m + 1 (also called antiprimitive) is studied. These codes are also linear codes with complementary duals (called LCD codes). Through some approaches such as iterative algorithm, partition and scaling, all coset leaders of C x modulo n with q ⌈ m 2 ⌉ < x ≤ 2 q ⌈ m 2 ⌉ + 2 are given for m ≥ 4. And for odd m the first several largest coset leaders modulo n are determined. Furthermore, a new kind of sequences is introduced to determine the second largest coset leader modulo n with m even and q odd. Also, for even m some conjectures about the first several coset leaders modulo n are proposed, whose complete verification would wipe out the difficult problem to determine the first several coset leaders of antiprimitive BCH codes. After deriving the cardinalities of the coset leaders, we shall calculate exact dimensions of many antiprimitive LCD BCH codes. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of q²-ary BCH codes with length n = q2m + 1 (also called antiprimitive BCH codes in the literature), where q ≥ 4 is a power of 2 and m ≥ 2. By a detailed analysis of some useful properties about q²-ary cyclotomic cosets modulo n, Hermitian dual-containing conditions for a family of non-narrowsense antiprimitive BCH codes are presented, which are similar to those of q²-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes. [ABSTRACT FROM AUTHOR]

BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths n , namely n = q m − 1 , n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 . For narrow-sense primitive BCH codes with designed distance δ , we investigate their dimensions for δ in the range 1 ≤ δ ≤ q ⌈ m 2 ⌉ + 1 . For non-narrow sense primitive BCH codes, we provide two general formulas on their dimensions and give the dimensions explicitly in some cases. Furthermore, we settle the minimum distances of some primitive BCH codes. We also explore the dimensions of the BCH codes of lengths n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 over finite fields. [ABSTRACT FROM AUTHOR]

Maximal-entanglement entanglement-assisted quantum error-correcting codes (EAQE-CCs) can achieve the EA-hashing bound asymptotically and a higher rate and/or better noise suppression capability may be achieved by exploiting maximal entanglement. In this paper, we discussed the construction of quaternary zero radical (ZR) codes of dimension five with length . Using the obtained quaternary ZR codes, we construct many maximal-entanglement EAQECCs with very good parameters. Almost all of these EAQECCs are better than those obtained in the literature, and some of these EAQECCs are optimal codes. [ABSTRACT FROM AUTHOR]

The increased use of biometric-based authentication systems in a variety of applications has made biometric template protection an important issue. Unlike conventional systems, biometric cannot be revoked or changed. This made template protection a critical issue to be considered in the recent years. This paper proposes a cancelable fingerprint cryptosystem using multiple spiral curves and fuzzy commitment scheme. The method is built by combining cancelable biometrics and biometric cryptosystems. First, we compute transformed minutiae features using multiple spiral curves. Further, these transformed features are encrypted using fuzzy commitment scheme. Hence, a secure template is obtained. Experimental results and analysis prove the credibility of proposed method with recently presented methods of fingerprint template protection. [ABSTRACT FROM AUTHOR]

This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field ( is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature. [ABSTRACT FROM AUTHOR]

The Rosenbloom–Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed–Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom–Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed–Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom–Tsfasman metric is provided. Two important properties of Reed–Solomon codes and BCH codes for the Rosenbloom–Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives. [ABSTRACT FROM PUBLISHER]

A free-space optical (FSO) communication link with multiplexed orbital angular momentum (OAM) modes has been demonstrated to largely enhance the system capacity without a corresponding increase in spectral bandwidth, but the performance of the link is unavoidably degraded by atmospheric turbulence (AT). In this paper, we propose a turbulence mitigation scheme to improve AT tolerance of the OAM-multiplexed FSO communication link using both channel coding and wavefront correction. In the scheme, we utilize a wavefront correction method to mitigate the phase distortion first, and then we use a channel code to further correct the errors in each OAM mode. The improvement of AT tolerance is discussed over the performance of the link with or without channel coding/wavefront correction. The results show that the bit error rate performance has been improved greatly. The detrimental effect of AT on the OAM-multiplexed FSO communication link could be removed by the proposed scheme even in the relatively strong turbulence regime, such as C n 2 = 3.6 × 10 − 14 m − 2 / 3 . [ABSTRACT FROM AUTHOR]

In this paper, we firstly study construction of new quantum error-correcting codes (QECCs) from three classes of quaternary imprimitive BCH codes. As a result, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing quaternary BCH codes are determined to be much larger than the result given according to Aly et al. (IEEE Trans Inf Theory 53:1183-1188, 2007) for each different code length. Thus, families of new QECCs are newly obtained, and the constructed QECCs have larger distance than those in the previous literature. Secondly, we apply a combinatorial construction to the imprimitive BCH codes with their corresponding primitive counterpart and construct many new linear quantum codes with good parameters, some of which have parameters exceeding the finite Gilbert-Varshamov bound for linear quantum codes. [ABSTRACT FROM AUTHOR]

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length $$n=q^{m}-1$$ over $$F_{q}$$ , where $$q\ge 5$$ is a prime power. The second one is the asymmetric quantum codes with length $$n=3^{m}-1$$ . These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set $$T_{1}=T_{2}^{-q}$$ , then the real Z-distance of our asymmetric quantum codes are much larger than $$\delta _\mathrm{max}+1$$ , where $$\delta _\mathrm{max}$$ is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature. [ABSTRACT FROM AUTHOR]

The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let C be an [ n , k ] cyclic code over F q , where F q is the finite field of order q. In this paper, we will employ the defining set of the code C to present a general characterization when its hull has dimension k − ℓ. Furthermore, we mainly focus on the primitive q -ary BCH codes C (q , n , δ , b) when b = 0 and b = 1 based on the general characterization. Especially for binary and ternary cases, we will present several sufficient and necessary conditions that the hulls of the codes C (q , n , δ , b) have dimensions k − 2 and k − 3 by giving lower and upper bounds on their designed distances, which extends the results of [17]. In addition, several classes of binary and ternary self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are investigated in some special cases. [ABSTRACT FROM AUTHOR]

• A watermarking method based on CGI and BCH error correction coding technique is proposed. • Binary images, grayscale images and even color images can be used as watermark. • The reconstructed watermark is lossless. • The scheme is more secure and robust due to the XOR operation and the use of BCH error correction code. The conventional image hiding scheme based on computational ghost imaging (CGI) has low-quality reconstructed watermark and unsatisfied security. In this paper, a high performance optical image hiding scheme is proposed to cope with these issues. The watermark is first read as a binary number sequence and processed by exclusive-OR operation with a random number sequence that generated by chaotic system. Thereafter, the result is coded with Bose-Chaudhuri-Hocquenghem error correction code and encrypted by the CGI system. Finally, the obtained cyphertext is quantified into two levels and hidden in the host image. Differing from other watermarking schemes, binary images, grayscale images and color images can be used as watermarks to be reconstructed losslessly, and a series of simulation results illustrate the feasibility and effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

For digital images, many watermarking schemes regarding copyright issue have been proposed. However, there exist many attacks trying to break or remove the embedded watermark. Therefore, the embedded watermark should be robust against all kinds of attacks. Among various attacks, the print and scan (PS) attack is a challenging one because it not only alters the pixel values but also changes the positions of original pixels. Most of the existing watermarking systems would fail under such an attack. In this paper, a watermarking system operating in the discrete cosine transform domain is proposed for colour images. By efficiently integrating the three components, red, green and blue, the proposed system exhibits superior robustness against various attacks, including the PS attack. According to the simulation results, the system maintains an acceptable image quality while providing outstanding robustness against the PS attack. [ABSTRACT FROM AUTHOR]

This paper presents a robust readable H.264/AVC data hiding algorithm without intra-frame distortion drift. We first divide the original embedded data into several groups by using the secret sharing technique. Then we use the BCH syndrome code (BCH code) technique to encode each grouping of data. Finally, we embed the encoded data into the paired-coefficients of the 4×4 discrete cosine transform (DCT) block of the selected frames which meet our conditions to avert the distortion drift. The experimental results show that this new data hiding algorithm can get more robustness, effectively avert intra-frame distortion drift and get high visual quality. [ABSTRACT FROM AUTHOR]

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present two constructions of irreducible (primitive) polynomials over F q of degree rm from irreducible (primitive) polynomial over F q m of degree r , and we show that these two constructions coincide. The first construction is based on the Frobenius automorphism of F q m over F q. The second one comes from a generalization of a construction of primitive polynomials over F q which uses the companion matrix. From this generalization, given an irreducible (resp. primitive) polynomial over F q m of degree r , we generate multiple (resp. all) irreducible polynomials over F q of degree rm. As an application, a characterization of the generator polynomial of a BCH code over F q is given. Then, we show how two BCH codes over F q and F q m , respectively, and their generator polynomials are related. [ABSTRACT FROM AUTHOR]

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over F q with designed distance δ = a q m − 1 − 1 (resp. δ = a q m − 1 q − 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 a q m − 1 q − 1 , a q m − 1 q − 1 + T for 0 ≤ a ≤ q − 2 , where T = q m + 1 2 − 1 if m is odd, and T = 2 q m 2 − 1 if m is even. [ABSTRACT FROM AUTHOR]

In the paper we investigate the structure of i -components of two classes of codes: the Kerdock codes and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code. We prove that for any admissible length the punctured Kerdock code consists of two i -components and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code is an i -component for any i. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restrictions of the Hamming scheme to the doubly shortened Kerdock codes are association schemes. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to investigate the parameters of some q -ary cyclic codes of length n = q 2 m − 1 q + 1. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented by employing the Hartmann-Tzeng bound or BCH bound. Moreover, when q = 2 , 3 , we give the largest coset leaders δ 1 modulo n and investigate the parameters of the BCH codes C (q , n , δ 1 , 1). [ABSTRACT FROM AUTHOR]

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths n=q2m-1q2-1 and n=q2m-1q+1 (m≥3) over the finite field Fq2 are studied. The maximum designed distances δmaxnew 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. [ABSTRACT FROM AUTHOR]