Let Fq be the finite field of q elements, where q = pm with p being a prime number and m being a positive integer. Let C(q,n,delta,h) be a class of BCH codes of length nand designed distance delta. A linear code Cis s...
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Let Fq be the finite field of q elements, where q = pm with p being a prime number and m being a positive integer. Let C(q,n,delta,h) be a class of BCH codes of length nand designed distance delta. A linear code Cis said to be maximum distance separable (MDS) if the minimum distance d = n - k + 1. If d = n - k, then Cis called an almost MDS (amds) code. Moreover, if both of C and its dual code C perpendicular to are amds, then Cis called a near MDS (NMDS) code. In [9], Geng, Yang, Zhang and Zhou proved that the BCH code C(q,q+1,3,4) is an almost MDS code, where q = 3m and m is an odd integer, and they also showed that its parameters is [q + 1, q - 3, 4]. Furthermore, they proposed a conjecture stating that the dual code C perpendicular to (q,q+1,3,4) is also an amds code with parameters [q + 1, 4, q - 3]. In this paper, we introduce the concept of subset code and use it together with the MacWilliams identity to establish characterizations for the dual code of an amds code to be an amds code. Then by this criteria, we show that the Geng-Yang-Zhang-Zhou conjecture is true. Our result together with the Geng-Yang-Zhang-Zhou theorem implies that the BCH code C(q,q+1,3,4) is an NMDS code. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we study constacyclic codes of length n = 7p(s) over a finite field of characteristics p, where p not equal 7 is an odd prime number and s a positive integer. The previous methods in the literature that...
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In this paper, we study constacyclic codes of length n = 7p(s) over a finite field of characteristics p, where p not equal 7 is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths np(s) with 1 <= n <= 6 cannot be applied to completely determine the Hamming distances of those with n = 7. This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length 7p(s). Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and amds codes among them will also be characterized.
Tang and Ding (IEEE Trans Inf Theory 67(1):244-254, 2021) opened a new direction of searching fort-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes s...
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Tang and Ding (IEEE Trans Inf Theory 67(1):244-254, 2021) opened a new direction of searching fort-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over GF(3(m)). Based on them, we present more infinite families of 3-designs that contain some 3-designs with new parameters as checked by Magma for small numbers m. We also construct an infinite family of cyclic codes over GF(q(2))and prove that the codewords of any nonzero weight support a 3-design. In particular, we present an infinite family of 6-dimensional amds codes over GF(3(2m))holding an infinite family of 3-designs and an infinite family of 7-dimensionalNMDS codes over GF(3(2m))holding an infinite family of 3-designs.
The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes ...
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The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length np(s) over F-pm, where p is a prime number and p does not divide n, the problem has been solved completely for all n <= 6 and partially for n = 7, 8. In this paper, we solve the problem for n = 9 and all primes p different from 3 and 19. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length 9p(s) over F-pm. As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes.
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