Some Results on the Weight Distributions of the Binary Double-Circulant Codes Based on Primes

Tjhai, Cen Jung; Tomlinson, M.; Horan, Robin; Ahmed, Mohammed; Ambroze, Marcel

doi:10.1109/iccs.2006.301431

Cited by 13 publications

(13 citation statements)

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“…Let p be a prime that is congruent to ±3 modulo 8. A binary [2(p + 1), p + 1] quadratic double-circulant code (QDC) [26], denoted by B, can be constructed using the following defining polynomials:…”

Section: Validation Of the Multiple Impulse Methodsmentioning

confidence: 99%

On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods

Mohamed¹,

Azouaoui²,

Nouh³

et al. 2012

IJCNS

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The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which we call "Multiple Impulse Method (MIM)", where the principle is to search codewords locally around the all-zero codeword perturbed by a minimum level of noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum Hamming-weight codeword whose Hamming weight is equal to the minimum distance of the linear code

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Section: Validation Of the Multiple Impulse Methodsmentioning

confidence: 99%

On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods

Mohamed¹,

Azouaoui²,

Nouh³

et al. 2012

IJCNS

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“…This form has been extensively studied before, and is usually referred to as bordered double circulant codes. It has been shown that P SL 2 (p) acts on these codes using the generating matrices above in previous work, such as [5] and [16]. Our proof is an alternate to those when p ≡ 3 (mod 8).…”

Section: Proof (Of Proposition 38)mentioning

confidence: 73%

The weight distribution of quasi-quadratic residue codes

Boston¹,

Hao²

2018

Advances in Mathematics of Communications

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1. Introduction. Quasi-quadratic residue codes (QQR codes) are a family of binary linear codes. They were first introduced by Bazzi and Mitter[2] as a quasi-cyclic code. Their work set foundations for the study of QQR codes. They discovered the relation between weights of a QQR code and number of points on hyperelliptic curves. Joyner [8] built upon this relation, and revealed the link between the QQR code and the famous Goppa's Conjecture in coding theory.We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when p ≡ 3 (mod 8).QQR codes are similar to quadratic residue codes. They are asymptotically rate half codes (exactly rate half when p ≡ 3 (mod 4)). Also, as we will show, P SL 2 (p) acts as automorphisms of the extended QQR codes in a similar way as of the extended quadratic residue codes. Furthermore, when p ≡ 7 (mod 8), we will show that the QQR code is equivalent to the even subcode of the corresponding quadratic residue code direct sum with itself, and therefore their weight enumerators have close relations.We will utilize the result that P SL 2 (p) acts on these codes to prove a new discovery about their weight polynomials, i.e. they are divisible by (x 2 + y 2 ) d−1 , where d is the corresponding minimum distance. The proof uses shadows of codes, a powerful tool to study weight polynomials. We also apply this idea to quadratic residue codes, and prove that their weight polynomials are divisible by (x + y) d , with d being the minimum distance.These results impose strong conditions on the weight polynomials of quadratic residue codes and QQR codes. Combining the divisibility result and Gleason's Theorem, we can derive an efficient algorithm to compute the weight polynomials of QQR codes. We also use these results to correct the existing computational

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“…We will call these codes double circulant. These codes are sometimes called pure double circulant to distinguish them from bordered double circulant which are not quasi-cyclic [14].…”

Section: Definitions and Notationmentioning

confidence: 99%