Negacyclic Codes for the Lee Metric

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

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Negacyclic codes of prime power length over the finite non-commutative chain ring 𝔽pm[u,𝜃] 〈u2〉

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500919 ◽

2021 ◽

pp. 2150091

Author(s):

Teeramet Inchaisri ◽

Jirayu Phuto ◽

Chakkrid Klin-Eam

Keyword(s):

Algebraic Structure ◽

Prime Power ◽

Nonzero Element ◽

Dual Code ◽

Constacyclic Codes ◽

Chain Ring ◽

Negacyclic Codes ◽

Ring Isomorphism ◽

One To One ◽

Negacyclic Code

In this paper, we focus on the algebraic structure of left negacyclic codes of length [Formula: see text] over the finite non-commutative chain ring [Formula: see text] where [Formula: see text] is an automorphism on [Formula: see text]. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] is a nonzero element of the field [Formula: see text] such that [Formula: see text].

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New Class of Codes, Correcting Errors in the Lee Metric

2019 XVI International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY) ◽

10.1109/redundancy48165.2019.9003337 ◽

2019 ◽

Author(s):

Viacheslav Davydov

Keyword(s):

New Class ◽

Lee Metric

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New EAQMDS codes constructed from negacyclic codes

Quantum Information Processing ◽

10.1007/s11128-020-02934-9 ◽

2020 ◽

Vol 19 (12) ◽

Author(s):

Wan Jiang ◽

Shixin Zhu ◽

Xiaojing Chen

Keyword(s):

Negacyclic Codes

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Structure of repeated-root constacyclic codes of length 8ℓmpn

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500505 ◽

2019 ◽

Vol 12 (04) ◽

pp. 1950050

Author(s):

Saroj Rani

Keyword(s):

Finite Field ◽

Linear Codes ◽

Important Class ◽

Constacyclic Codes ◽

Dual Codes ◽

Negacyclic Codes ◽

Positive Integers

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that [Formula: see text] is the finite field of order [Formula: see text] where [Formula: see text] is a power of the prime [Formula: see text] and [Formula: see text] are distinct odd primes, and [Formula: see text] are positive integers. We determine generator polynomials of all constacyclic codes of length [Formula: see text] over the finite field [Formula: see text] We also determine their dual codes.

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