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The parameters of projective Reed–Müller codes

Gilles Lachaud
1990 Discrete Mathematics  
The projective Reed-Mtiller codes on a finite field are extensions of the classical generalized Reed-Mtiller codes.  ...  We give the parameters of these codes; we use a bound given by Serre in order to get their minimal distance.  ...  The weights of the projective Reed-Miiller code %&(r, Pm) satisfy w=O(modqb), Theorem 4 is the analog for projective Reed-Miiller codes in any degree and any characteristic of the Corollary 13 in Chapter  ... 
doi:10.1016/0012-365x(90)90155-b fatcat:j3jyvj6m3jce5bcpcqup7uay54

Projective reed-muller codes [chapter]

Gilles Lachaud
1988 Lecture Notes in Computer Science  
On donne les traram&res des codes de Reed-Muller projecfifs d'ordre 2. 1.Introduction We note pm(Fq) the projective space of dimension m over the finite field Fq with q elements, so that # pm(Fq) = qm  ...  On inlroduit une classe de codes lintaires de la famille des codes de Reed-Muller, les codes de Reed-Muller projectifs.  ...  Projective Reed-Muller Codes of order 1 : The Plotkin bound. For r = 1, the space Fq[X 0 .....  ... 
doi:10.1007/3-540-19368-5_13 fatcat:pbl4ssczvba2bn2v2pxvqw6d7u

Projected subcodes of the second order binary Reed-Muller code

Matthieu Legeay, Pierre Loidreau
2012 2012 IEEE International Symposium on Information Theory Proceedings  
The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code.  ...  In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters.  ...  We show that it is possible to project the second order Reed-Muller code onto many codes with smaller parameters -typically of the order of the parameters of the first order Reed-Muller code.  ... 
doi:10.1109/isit.2012.6283977 dblp:conf/isit/LegeayL12 fatcat:u5hzjpxg3jftbcfbbngbuyvwzy

The weight distribution of the self-dual $[128,64]$ polarity design code

Vladimir D. Tonchev, Ethan Novak, Masaaki Harada
2016 Advances in Mathematics of Communications  
The weight distribution of the binary self-dual [128,64] code being the extended code C^* of the code C spanned by the incidence vectors of the blocks of the polarity design in PG(6,2) [11] is computed  ...  Acknowledgments The authors thank the anonymous reviewers for the useful comments and suggestions that lead to some improvements of the text.  ...  distribution of the 3rd order Reed-Muller code R(3, 7).  ... 
doi:10.3934/amc.2016032 fatcat:gmqtqgiourfdlj4vb7irqyqoku

Reed-Muller codes associated to projective algebraic varieties [chapter]

Yves Aubry
1992 Lecture notes in mathematics  
If V is of degree h, we give parameters of Reed-Muller codes of order d < h, associated to V.  ...  In application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters  ...  determine the parameters of the projective Reed-Muller codes of order 1 associated to Q.  ... 
doi:10.1007/bfb0087988 fatcat:ojnkehlrhfagxbdhfujnclbu7a

Page 5267 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
The parameters of generalised Reed-Muller codes described using the algebraic geometrical approach are generalised by means of weighted Reed-Muller codes.  ...  The affine and projective geometrical versions of weighted Reed-Muller codes, referred to as WARM- and WPRM-codes, are investigated and their parameters determined.  ... 

Reed-Muller codes: projections onto GF(4) and multilevel construction

O. Amrani, Y. Be'ery
2001 IEEE Transactions on Information Theory  
A projection of binary Reed-Muller codes ( ) onto GF (4) is presented.  ...  This construction of Reed-Muller codes is readily applicable for their efficient decoding.  ...  Lahtonen for his insightful comments that greatly improved the presentation of the results.  ... 
doi:10.1109/18.945270 fatcat:zstifv6lrnfpzax7puou6qfgwu

Direct products in projective Segre codes [article]

Azucena Tochimani, Maria Vaz Pinto, Rafael H. Villarreal
2016 arXiv   pre-print
We introduce a family of projective Reed-Muller-type codes called projective Segre codes.  ...  Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed-Muller-type codes.  ...  We thank the referees for their careful reading of the paper and for the improvements that they suggested.  ... 
arXiv:1501.01692v4 fatcat:dqvruxpezvfebk3ur67xzhowki

On the generalized Hamming weights of certain Reed–Muller-type codes

Manuel González-Sarabia, Delio Jaramillo, Rafael H. Villarreal
2020 Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica  
parameters and the generalized Hamming weights of the corresponding projective ReedMuller-type codes on 𝕏 and their dual codes.  ...  If 𝕏 is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on 𝕏 and their dual codes in terms of the basic  ...  If X = P 2 , then by Theorem 3.3 the basic parameters of the classical projective Reed-Muller-type code C X (d) of degree d are given by  ... 
doi:10.2478/auom-2020-0014 fatcat:fxsa2ngypreebet4pkqsx4here

On the generalized Hamming weights of certain Reed-Muller-type codes [article]

Manuel Gonzalez-Sarabia, Delio Jaramillo, Rafael H. Villarreal
2019 arXiv   pre-print
and the generalized Hamming weights of the corresponding projective ReedMuller-type codes on X and their dual codes.  ...  If X is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on X and their dual codes in terms of the basic parameters  ...  Veronese codes are a natural generalization of the classical projective Reed-Muller codes.  ... 
arXiv:1905.12136v2 fatcat:q7dt5si4ofg4lnlww4reqy6uxe

Group Structure on Projective Spaces and Cyclic Codes over Finite Fields

Gilles Lachaud, Isabelle Lucien, Dany-Jack Mercier, Robert Rolland
2000 Finite Fields and Their Applications  
The geometrical properties of these groups give a fairly simple description of these codes which are of the Reed}Muller kind. Academic Press  ...  We construct some cyclic codes related to Reed}Muller codes by evaluating polynomials on these subgroups.  ...  Hence the parameters of these two codes can be derived from the known parameters of the projective Reed}Muller code (cf. [5, 8] ). (c) Let G"G L\ "+x " N(x)"1, and m"gcd(n, q!1).  ... 
doi:10.1006/ffta.1999.0268 fatcat:imbme4pcrngelheqisy72bso3u

On the weight distribution of second order Reed-Muller codes and their relatives [article]

Shuxing Li
2019 arXiv   pre-print
In addition, the weight distributions of second order q-ary homogeneous Reed-Muller codes and second order q-ary projective Reed-Muller codes are also determined.  ...  Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provide a precise account for the weight distribution of second order q-ary Reed-Muller codes.  ...  Consequently, we complete the proof. Employing Lemma 4.2 and comparing Definition 2.3 with [5, Proposition 3.8], we confirm the correspondence in Table 2 .1.  ... 
arXiv:1903.08058v1 fatcat:3elitsg7hvcqxgbucprie6w6fq

Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

Vinayak Ramkumar, Myna Vajha, P. Vijay Kumar
2018 2018 Twenty Fourth National Conference on Communications (NCC)  
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous.  ...  In prior work [1] by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes.  ...  The projective Reed-Muller (PRM) codes, introduced by Gilles Lachaud in [7] , are a variant of the Reed-Muller (RM) codes.  ... 
doi:10.1109/ncc.2018.8600134 fatcat:asjdg56b4fghfiytq2p3lr2fda

Page 5349 of Mathematical Reviews Vol. , Issue 89I [page]

1989 Mathematical Reviews  
Using projective space instead of affine space the author generalizes the construction of generalized Reed-Muller codes to a class of codes he calls projective Reed-Muller codes.  ...  He gives the exact value of the minimum weight for second-order projective Reed-Muller codes. {For the entire collection see MR 89c:94004.}  ... 

On the Local Correctabilities of Projective Reed-Muller Codes [article]

Sian-Jheng Lin
2017 arXiv   pre-print
In this paper, we show that the projective Reed-Muller (PRM) codes form a family of locally correctable codes (LCC) in the regime of low query complexities.  ...  This paper shows that PRM codes are shorter than generalized Reed-Muller (GRM) codes in LCCs.  ...  REED-MULLER CODES AND PROJECTIVE REED-MULLER CODES Codes Restriction Query complexity Message length Code length Minimum distance GRM d ≤ q − 2 d + 1 m+d d  ... 
arXiv:1702.02671v1 fatcat:llrs3lruv5cu7docylqi2rha4y
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