On Hamming and b -symbol distance distributions of repeated-root constacyclic codes of length $$4p^s$$ 4 p s over $${
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On Hamming and b-symbol distance distributions of repeated-root constacyclic codes of length 4ps over Fpm + uF F pm Hai Q. Dinh1,2 · Abhay Kumar Singh3 · Madhu Kant Thakur3 Received: 15 July 2020 / Revised: 24 October 2020 / Accepted: 24 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Let p be a prime such that p m ≡ 1 (mod 4), and R = F pm + uF pm . For any nonsquare unit λ of R, the Hamming and b-symbol distances of all λ-constacyclic codes of length 4 p s over R are completely determined. As examples, several good codes with new parameters are constructed. We also identified all Maximum Distance Separable constacyclic codes of length 4 p s over R with respect to the Hamming distance as well as the b-symbol distance. Also, we got some non-trivial MDS b-symbol Type 3, γ -constacyclic codes of length 4 p s codes over R with respect to b-symbol distance for b = 4. Keywords Finite chain rings · Constacyclic codes · Repeated-root codes · Hamming distance · b-symbol distance · MDS codes Mathematics Subject Classification 94B15 · 11T71
1 Introduction Theoretically, constacyclic codes are the pivotal and profound part of linear codes. Constacyclic codes are generalization of cyclic codes, which form the most important
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Madhu Kant Thakur [email protected] Hai Q. Dinh [email protected] Abhay Kumar Singh [email protected]
1
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
3
Department of Mathematics and Computing, Indian Institute of Technology (ISM), Dhanbad, India
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and well studied class of error-correcting codes. Due to their encoding with shift registers, these codes have numerous practical applications. In recent years, there has been a great interest in study about the algebraic structure of constacyclic codes and apply to construct good codes. For a unit λ of F pm , λ-constacyclic codes of length n over F pm are ideals of the F
m [x]
ring Rλ = xpn −λ . The constacyclic codes of length n are said to be simple-root constacyclic codes if gcd(n, p) = 1. Otherwise, the constacyclic codes are said to be repeated-root constacyclic codes. In [1], Berman initiated the study of repeated-root constacyclic codes. Afterward, many researchers studied repeated-root constacyclic codes over finite fields [3,6–8,24,25] and finite chain rings [4,5,10,11,15,16,21,22]. However, till now very little amount of works on computation of the Hamming distances have been done due to computational complexity. In [4], Dinh obtained the Hamming distances of all the cyclic codes of prime power lengths over F pm . Later, in [17], Dinh et al. computed the Hamming distances of all constacyclic codes of length 4 p s over F pm . In [5], Dinh determined Hamming distances of all (α + uβ)-constacyclic codes of length of prime power over R. Later, in [12], Din
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