Negacyclic bchcodes form an important subclass of negacyclic codes and can produce optimal linear codes in many cases. The question of whether the dual code of a negacyclic bchcode is a negacyclic bchcode is, in ge...
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Negacyclic bchcodes form an important subclass of negacyclic codes and can produce optimal linear codes in many cases. The question of whether the dual code of a negacyclic bchcode is a negacyclic bchcode is, in general, very hard to answer. To investigate further the properties of the dual codes of negacyclic bchcodes, the concept of negacyclic duallybchcodes is proposed in this paper and then the dual codes of narrow-sense negacyclic bchcodes of length qm+1 2 over the finite field Fq are studied, where q = 3 (mod 4). Some lower bounds on the minimum distances of the dual codes are established, which are very close to the true minimum distances of the dual codes in many cases. Sufficient and necessary conditions in terms of designed distances are presented to ensure that narrow-sense negacyclic bchcodes of length qm+1 2 are negacyclic dually-bch codes. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
bchcodes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of bch cyclic codes has been made, but little is known about the mi...
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bchcodes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of bch cyclic codes has been made, but little is known about the minimum distances of duals of bchcodes. Recently, a concept called dually-bch code was introduced to investigate the duals of bchcodes and the lower bounds on their minimum distances in Gong et al., (2022). For a prime power q and an integer m >= 4, let n = q(m)-1 /q+1 (m even), or n = q(m)-1/ q-1 (q > 2). In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense bchcodes of length n are dually-bch codes, which extended the results in Gong et al., (2022). Lower bounds on the minimum distances of their dual codes are developed for n = q(m)-1 /q+1 (m even). As byproducts, we present the largest coset leader delta(1) modulo n being of two types, which proves a conjecture in Wu et al., (2019) and partially solves an open problem in Li et al., (2017). We also investigate the parameters of narrow-sense bchcodes of length n with design distance delta(1). The bchcodes presented in this paper have good parameters in general.
作者：
Wang, XiaoqiangXiao, ChengliangZheng, DabinHubei Univ
Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Peoples R China Hubei Univ
Fac Math & Stat Hubei Key Lab Appl Math Minist Educ Wuhan 430062 Peoples R China Hubei Univ
Key Lab Intelligent Sensing Syst & Secur Minist Educ Wuhan 430062 Peoples R China
bchcodes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of bchcodes has been made, but little is known about...
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bchcodes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of bchcodes has been made, but little is known about the properties of their duals. Recently, in order to study the duals of bchcodes and the lower bounds on their minimum distances, a new concept called dually-bch code was proposed by (Gong et al., 2022). In this paper, the lower bounds on the minimum distances of the m - 1 duals of narrow-sense bchcodes with length q lambda over Fq are developed, where lambda is a positive integer satisfying lambda = q s - 1 and s m , or lambda q - 1. In addition, the sufficient and necessary conditions in terms of the designed distances for these codes being dually-bch codes are presented. Our lower bounds on the minimum distances of the duals of bchcodes include the bounds stated in (Gong et al., 2022) as a special case. Moreover, our lower bounds improve the bounds stated in (Gong et al., 2022), the classical Sidel'nikov bound, and the Carlitz-Uchiyama bound when the designed distances of the bchcodes are in some ranges. Several examples show that our proposed lower bounds are good in some cases.
bchcodes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of bchcodes, however, has been an important but difficult problem. Recently, in orde...
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bchcodes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of bchcodes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of bchcodes, the concept of dually-bch codes was proposed. In this paper, we study bchcodes 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 bchcodes of length q(m) +1/q +1 with designed distance delta = lq(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 bchcodes 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 bchcodes of length q(m) +1 being dually-bch codes are presented, where q is odd and m (sic) 0 (mod 4). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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