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As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length q m + 1 λ where λ ∣ q + 1 .Some new techniques are employed to study the coset leaders. For any odd prime power q and m = 4 , 8 , or m ≥ 12 and m ≡ 4 (mod 8) , the second, the third and the fourth largest coset leaders modulo q m + 1 are determined, and the dimensions of some BCH codes of length q m + 1 with large designed distances are given. For 1 < λ < q + 1 , the first few largest coset leaders and the coset leaders modulo q m + 1 λ in the range 1 to q ⌊ (m + 1) / 2 ⌋ λ are studied, and the dimensions of some BCH codes of length q m + 1 λ are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound. [ABSTRACT FROM AUTHOR]

Telecommand (TC) plays a crucial role in the success of high-altitude balloon experiments. Bose, Ray-Chaudhuri, Hocquenghem (BCH) codes are commonly employed to ensure reliable command operation. The Balloon Facility (BF) of Tata Institute of Fundamental Research (TIFR) uses a TC system based on BCH (31,16) coding technique, to control balloon and payload operations. This paper presents prototyping and implementation of TC encoder and decoder using Spartan 6 Field Programmable Gate Array (FPGA). The code is written in Very high-speed integrated circuit Hardware Description Language (VHDL). Simulation and synthesis are done using Xilinx ISE 14.7 design suite. Simulation results show the design is robust. The TC encoder is implemented in a commercial FPGA development board and the TC decoder is implemented in a specially designed FPGA board, successfully. This paper presents the salient features of the TC system in use and the implementation of the system using FPGA. [ABSTRACT FROM AUTHOR]

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]

Negacyclic BCH codes are a generalization of BCH codes, they have better parameters than BCH codes in some cases. In this paper, we investigate negacyclic BCH codes of length q m + 1 2 and q m - 1 2 λ over F q , where q ≥ 3 is an odd prime power and λ is an integer with 2 ≤ λ ≤ q - 1 , λ ∣ (q - 1) and 2 λ ∣ (q m - 1) . By using some results on the q-cyclotomic cosets modulo q m + 1 or q m - 1 λ , we determine the dimensions of these two classes of negacyclic BCH codes, and give some lower bounds on the minimum distance of these codes. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals (LCD codes) have attracted a lot of interest in recent years because of their applications in implementations against side-channel attacks and fault injection attacks. In this paper, we investigate the parameters of ternary LCD constacyclic BCH codes of lengths 3 m - 1 / 2 and 3 m + 1 / 2. For length n = 3 m - 1 / 2 , the parameters of LCD negacyclic BCH codes are explored by calculating the first several largest odd absolute coset leaders modulo 3 m - 1 , and the minimum distance of a class of such LCD codes are settled. For length n = 3 m + 1 / 2 , the parameters of ternary LCD cyclic BCH codes with designed distance 2 ≤ δ ≤ 3 m / 2 + 2 are discussed by tracing coset leaders modulo 3 m + 1 / 2. Some of these BCH codes that are LCD with exact minimum distance are presented. [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]

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]

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]

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]

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]

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]

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 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]

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]

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]

Digital filters are increasingly being used in signal processing areas. The soft errors present in these circuits are found to affect the reliability of the systems such as biomedical and space applications. In this work, fault-tolerant digital finite impulse response (FIR) filters are designed to have the same impulse response that process different input signals and multiple responses with constant input signals using the soft computing approaches. The soft computing approaches used in the proposed FIR filters are Bose–Chaudhuri–Hocquenghem (BCH) code for single error correction and double error correction. The performance of the proposed method is evaluated by considering different FIR filter tap configurations. The Questasim simulator has been used to validate the functionality of the designed modules using a Verilog HDL. The Xilinx Vivado HLx 2018.3 and Artix-7 xc7a200 and Virtex-4 XC4VLX80 FPGA devices have been used for synthesis and implementation. Significant savings up to 28.02% and 21.8%, 26.67% and 26.33% of different field programmable gate array (FPGA) architecture resources like Slices, LUTs, flip-flops, and LUTRAMs for the parallel filters having the same impulse response and different impulse responses, respectively, are achieved as compared to the triple modular redundancy method. The digital parallel FIR filters implemented with FPGA shows effectiveness in terms of correction of single and double errors by employing the BCH code, implementation cost, and protection. [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]

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]