Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Niu, Yongfeng; Yue, Qin; Wu, Yansheng; Hu, Liqin

doi:10.1109/lcomm.2019.2908640

Cited by 21 publications

(14 citation statements)

References 16 publications

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“…The method here used basically the use of MATLAB and in the design, the main component needed is the RS encoder and decoder [15]. The idea was to implement by using the Simulink in order to make it better and faster to implement, because when you use reed Solomon codes it will take a while and will consume a lot of data unlike, if its Simulink the idea is just to place block that represents the reed Solomon and it understood in that in that block of reed Solomon encoder or decoder, the Reed Solomon code is already embedded inside the block.…”

Section: Methodsmentioning

confidence: 99%

Development of a Reed-Solomon Error Detection and Correction System

Africa¹

2020

IJETER

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This project scopes into the Reed-Solomon (RS) Code as one of the optimal error detection and correction schemes. RS codes are codes in which adding unnecessary bits to a data in order for it to be reliably recovered even though the presence of errors in transferal, cache, and retrieval, is executed. The encoder for this specific code differs from a binary encoder in that it operates on the multiple bits rather than individual bits giving it the ability to precisely detect and correct error compared to other error and correction schemes. Its unique algorithm may be used in any analog or digital communication system because of it having the highest efficient use of redundancy, ability to adjust block length has a wide range of code rate and is an efficient coding technique perfect for accurate and precise code error detection. Storage devices, wireless communication, satellite communication, digital television, and high-speed modems are constructs of Reed-Solomon.

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Section: Methodsmentioning

confidence: 99%

Development of a Reed-Solomon Error Detection and Correction System

Africa¹

2020

IJETER

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“…Using Algorithm 14 with an optimal non-SO [15,5,7] code, we construct a Reed-Muller [16,5,8] SO code with generator matrix 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 , whose dual code is a [16,11,4] code. Thus, by Corollary 23, we obtain an optimal [ [16,6,4]] quantum code by [6]. In the same manner, we obtain [ [11,3,3]] and [ [15,7,3]] quantum codes which are optimal by [6].…”

Section: Optimal Self-orthogonal Codes Of Dimensionmentioning

confidence: 99%

“…One of the main topics in coding theory is to find minimum distance optimal code among self-dual or self-orthogonal codes [16]. Boukllieve et al [2] investigated optimal [ , ] SO codes of lengths for ≤ 40 and ≤ 10.…”

Section: Introductionmentioning

confidence: 99%

Self-orthogonality matrix and Reed-Muller code

Kim¹,

Choi²

2021

Preprint

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Kim et al. (2021) gave a method to embed a given binary [ , ] code C ( = 3, 4) into a self-orthogonal code of the shortest length which has the same dimension and minimum distance ′ ≥ (C). We extends this result for = 5 and 6 by proposing a new method related to a special matrix, called the self-orthogonality matrix , obtained by shortnening a Reed-Muller code R (2, ). Furthermore, we disprove partially the conjecture (Kim et al. (2021)) by showing that if 31 ≤ ≤ 256 and ≡ 14, 22, 29 (mod 31), then there exist optimal [ , 5] codes which are self-orthogonal. We also construct optimal self-orthogonal [ , 6] codes when 41 ≤ ≤ 256 satisfies

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“…Using the above theorem, numerous MDS Hermitian self-dual codes over F q + uF q + u 2 F q can be construct based on known MDS Hermitian self-dual codes over F q (see, for example, [13], [17], [24]).…”

Section: Hermitian Self-dual Linear Codes Overmentioning

confidence: 99%

Self-dual codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and applications

Choosuwan

Jitman

2020

Journal of Algebra Combinatorics Discrete Structures and Applications

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Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring Fq + uFq + u 2 Fq with u 3 = 0 have been established. In this paper, Hermitian self-dual linear codes over Fq + uFq + u 2 Fq are studied for all square prime powers q. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of H-quasi-abelian codes in Fq[G] is studied, where H ≤ G are finite abelian groups and Fq[H] is a principal ideal group algebra. General characterization and enumeration of H-quasi-abelian codes and self-dual H-quasi-abelian codes in Fq[G] are given. For the special case where the field characteristic is 3, an explicit formula for the number of self-dual A × Z3-quasi-abelian codes in F3m [A × Z3 × B] is determined for all finite abelian groups A and B such that 3 |A| as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over F3m + uF3m + u 2 F3m . Some illustrative examples are provided as well.

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Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Cited by 21 publications

References 16 publications

Development of a Reed-Solomon Error Detection and Correction System

Development of a Reed-Solomon Error Detection and Correction System

Self-orthogonality matrix and Reed-Muller code

Self-dual codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and applications

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