CYCLOTOMIC COSETS MODULO m

Cyclotomic classes of order 2 with respect to a product of two distinct odd primes [Formula: see text] and [Formula: see text] are represented in some specific forms and using these forms an alternate proof of Theorem 3 of [C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl. 4 (1998) 140–166] is given, when [Formula: see text]. Further, it is observed that these classes are related to [Formula: see text]-cyclotomic cosets, where [Formula: see text] and [Formula: see text] such that gcd([Formula: see text]. Finally, arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are studied.

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Four classes of linear codes from cyclotomic cosets

Designs Codes and Cryptography ◽

10.1007/s10623-017-0374-0 ◽

2017 ◽

Vol 86 (5) ◽

pp. 1007-1022 ◽

Cited By ~ 3

Author(s):

Dabin Zheng ◽

Jingjun Bao

Keyword(s):

Linear Codes ◽

Cyclotomic Cosets

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Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

International Journal of Quantum Information ◽

10.1142/s0219749914500154 ◽

2014 ◽

Vol 12 (03) ◽

pp. 1450015 ◽

Cited By ~ 22

Author(s):

Liang-Dong Lü ◽

Ruihu Li

Keyword(s):

Linear Codes ◽

Careful Analysis ◽

Galois Field ◽

Error Correcting Codes ◽

Optimal Number ◽

Bch Codes ◽

Design Parameters ◽

Bch Code ◽

Cyclotomic Cosets ◽

Quantum Error

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

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Structured quasi-cyclic low-density parity-check codes based on cyclotomic cosets

IET Communications ◽

10.1049/iet-com.2014.0236 ◽

2015 ◽

Vol 9 (4) ◽

pp. 541-547 ◽

Cited By ~ 2

Author(s):

Morteza Esmaeili ◽

Mehrab Najafian ◽

Aaron T. Gulliver

Keyword(s):

Low Density ◽

Parity Check ◽

Cyclotomic Cosets ◽

Low Density Parity Check ◽

Parity Check Codes

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Cyclotomic Cosets, the Mattson–Solomon Polynomial, Idempotents and Cyclic Codes

Error-Correction Coding and Decoding - Signals and Communication Technology ◽

10.1007/978-3-319-51103-0_4 ◽

2017 ◽

pp. 61-99

Author(s):

Martin Tomlinson ◽

Cen Jung Tjhai ◽

Marcel A. Ambroze ◽

Mohammed Ahmed ◽

Mubarak Jibril

Keyword(s):

Cyclic Codes ◽

Cyclotomic Cosets

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New quantum constacyclic codes with length n=2(qm+1)

International Journal of Quantum Information ◽

10.1142/s0219749919500576 ◽

2019 ◽

Vol 17 (07) ◽

pp. 1950057

Author(s):

Junli Wang ◽

Ruihu Li ◽

Yang Liu ◽

Hao Song

Keyword(s):

Prime Power ◽

Bch Codes ◽

Quantum Codes ◽

Constacyclic Codes ◽

Cyclotomic Cosets ◽

The Given

By studying the properties of [Formula: see text]-cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length [Formula: see text] are determined, where [Formula: see text] is an odd prime power and [Formula: see text] is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature.

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