Constacyclic codes, cocycles and a u+v|u-v construction

Hughes, G.

doi:10.1109/18.825841

Cited by 36 publications

(20 citation statements)

References 9 publications

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“…Thus for the constacyclic codes n-isometric to cyclic codes, the following consequence is closely related to [42 q . The λ-constacyclic codes of length n are isometric to the cyclic codes of length n if and only if a n λ = 1 for an element a ∈ F * q ; further, in that case the map…”

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confidence: 95%

Recent progress on weight distributions of cyclic codes over finite fields

Dinh¹,

Li²,

Yue³

2015

JACODESMATH

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Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

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confidence: 95%

Recent progress on weight distributions of cyclic codes over finite fields

Dinh¹,

Li²,

Yue³

2015

JACODESMATH

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“…More generally, it was conjectured in [5, Section IV.D] that for any prime p a family ('Main-Sequence') MS p codes over F p having properties similar to those of the binary RM-codes' would be obtained using a construction with associated upper-triangular p × p F p -matrix MS p = p−i j −1 mod p . A (u + v|u − v)-construction (see [11, Theorem 6]) has also been applied to construct new codes in [4] and generalizations of Turyn's (a+x|b+x|a+b+x)-construction appear in a study of quasi-cyclic codes in [6].…”

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confidence: 99%

Matrix-Product Codes over ? q

Blackmore¹,

Norton²

2001

AAECC

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Codes C 1 , . . . , C M of length n over F q and an M × N matrix A over F q define a matrix-product codeWe study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ⊥ . It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be 'non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary 'Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ⊥ can be described using C ⊥ 1 , . . . , C ⊥ M and a transformation of A. This yields d(C ⊥ ). Finally we show that an NSC matrix-product code is a generalized concatenated code.

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“…The idempotent generators of minimal constacyclic codes are denoted by θ t (x) and those of the maximal constacyclic codes by ϑ t (x), t ∈ T n,q,λ [16,22,25]. The polynomials θ t (x) are often called primitive idempotent polynomials, since any idempotent generator can be written as a linear combination of these polynomials for fixed values of n, q and λ. Constacyclic codes with special parameter values or constacyclic codes constructed by special methods are discussed in [9,11,12,14,15].…”

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confidence: 99%

Primitive idempotent tables of cyclic and constacyclic codes

Zanten

2018

Des. Codes Cryptogr.

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Constacyclic codes, cocycles and a u+v|u-v construction

Cited by 36 publications

References 9 publications

Recent progress on weight distributions of cyclic codes over finite fields

Recent progress on weight distributions of cyclic codes over finite fields

Matrix-Product Codes over ? q

Primitive idempotent tables of cyclic and constacyclic codes

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