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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]

BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over G F (q) of length 2 (q 2 m − 1) q + 1 with small and large dimensions. We study the q -cyclotomic cosets modulo 2 (q 2 m − 1) q + 1 , determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual 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]

In this paper, narrow-sense and non-narrow-sense BCH codes of length n =2(q4−1) that contain their Hermitian duals over Fq2 are studied, where q is an odd prime power. Theory of cyclotomic coset is applied to confirm the maximum designed distances and dimensions of these BCH codes. Finally, some new quantum codes are obtained from such BCH codes, many of our quantum codes are much better than those from narrow-sense BCH codes in [12]. [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]

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]

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]

• 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]

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]

Abstract: In this paper, an optical security system has been proposed in which key of the double random phase encoding technique is linked to the biometrics of the user to make it user specific. The error in recognition due to the biometric variation is corrected by encoding the key using the BCH code. A user specific shuffling key is used to increase the separation between genuine and impostor Hamming distance distribution. This shuffling key is then further secured using the RSA public key encryption to enhance the security of the system. XOR operation is performed between the encoded key and the feature vector obtained from the biometrics. The RSA encoded shuffling key and the data obtained from the XOR operation are stored into a token. The main advantage of the present technique is that the key retrieval is possible only in the simultaneous presence of the token and the biometrics of the user which not only authenticates the presence of the original input but also secures the key of the system. Computational experiments showed the effectiveness of the proposed technique for key retrieval in the decryption process by using the live biometrics of the user. [Copyright &y& Elsevier]

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]