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On the next-to-minimal weight of projective Reed-Muller codes [article]

Cícero Carvalho, Victor G.L. Neumann
2017 arXiv   pre-print
In this paper we present several values for the next-to-minimal weights of projective Reed-Muller codes.  ...  We work over F_q with q ≥ 3 since in IEEE-IT 62(11) p. 6300-6303 (2016) we have determined the complete values for the next-to-minimal weights of binary projective Reed-Muller codes.  ...  One may wonder if there is a similar relation between the next-to-minimal weight of projective Reed-Muller codes and the next-to-minimal weight of generalized Reed-Muller codes.  ... 
arXiv:1701.01663v1 fatcat:2tmu6ir4ura7lpherrk6csdhuy

The Next-to-Minimal Weights of Binary Projective Reed–Muller Codes

Cicero Carvalho, Victor G. L. Neumann
2016 IEEE Transactions on Information Theory  
In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most  ...  Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined.  ...  Interestingly, we note that the next-to-minimal weight of the binary projective Reed-Muller codes of order d is equal to the next-to-minimal weight of the Reed-Muller codes of order d − 1 in most but not  ... 
doi:10.1109/tit.2016.2611527 fatcat:acqkv7hhp5etjof3p24bmq4fia

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.  ...  However, next-to-minimal weight of binary PRM codes is determined in a recent work [12] , wherein next-to-minimal weight means minimal codeword weight that is greater than minimum Hamming weight.  ... 
doi:10.1109/ncc.2018.8600134 fatcat:asjdg56b4fghfiytq2p3lr2fda

Page 9449 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
Summary: “A projection of binary Reed-Muller codes &(r, m) onto GF(4)”~? is presented.  ...  These lineav codes have parameters identical to those of binary Reed-Muller codes and some of their properties are also similar to those of RM codes.  ... 

Page 4099 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
The projective Reed-Muller codes on a finite field are extensions of the classical generalized Reed-Muller codes.  ...  The proposed codes are compared with classical Reed-Muller codes. Congruences of weights of projective Reed-Muller codes are derived from Ax’s theorem. Simon N.  ... 

Pure Resolutions, Linear Codes, and Betti Numbers [article]

Sudhir R. Ghorpade, Prasant Singh
2020 arXiv   pre-print
The nature of the minimal free resolution of Stanley-Reisner rings corresponding to several classes of two-weight codes, besides the first order Reed-Muller codes, is also determined.  ...  We use this characterization to quickly deduce the minimal free resolutions of Stanley-Reisner rings associated to MDS codes as well as constant weight codes.  ...  The authors are also grateful to the Indo-Norwegian project MAIT: EECC, supported by the Research Council of Norway and the Dept. of Science and Technology of Govt. of India, which facilitated mutual visits  ... 
arXiv:2002.01799v2 fatcat:i3fkzxz4mvho5ma3txpad67kpe

Page 5115 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
of the projective generalized Reed-Muller codes spanned by minimum-weight vectors.  ...  London, London, 1977; per bibl.] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals.  ... 

Codes over a weighted torus

Eduardo Dias, Jorge Neves
2015 Finite Fields and Their Applications  
We define the notion of weighted projective Reed-Muller codes over a subset X ⊂ P(w 1 , . . . , w s ) of a weighted projective space over a finite field.  ...  We compute the basic parameters of weighted projective Reed-Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.  ...  In Section 3 we define weighted projective Reed-Muller codes. We compute the length of the weighted projective Reed-Muller codes over the weighted torus (cf.  ... 
doi:10.1016/j.ffa.2014.11.009 fatcat:aiwi2vi5zjghnijb5hil2xd4by

Codes over a weighted torus [article]

Eduardo Dias, Jorge Neves
2013 arXiv   pre-print
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus.  ...  We compute the basic parameters of weighted projective Reed-Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.  ...  In Section 3 we define weighted projective Reed-Muller codes. We compute the length of the weighted projective Reed-Muller codes over the weighted torus (cf.  ... 
arXiv:1307.6380v1 fatcat:bdcyjtznxfejviiph467anudha

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  
We construct some cyclic codes related to Reed}Muller codes by evaluating polynomials on these subgroups.  ...  The geometrical properties of these groups give a fairly simple description of these codes which are of the Reed}Muller kind. Academic Press  ...  1)"1 then the subgroup G L\ has one and only one point on each punctured vector line of %* q L . Therefore the code RM(H P , G L\ ) is a generalized projective Reed}Muller code.  ... 
doi:10.1006/ffta.1999.0268 fatcat:imbme4pcrngelheqisy72bso3u

On low weight codewords of generalized affine and projective Reed–Muller codes

Stéphane Ballet, Robert Rolland
2014 Designs, Codes and Cryptography  
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes.  ...  In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.  ...  INTRODUCTION -NOTATIONS This paper proposes a study on low weight codewords of generalized Reed-Muller codes and projective generalized Reed-Muller codes of degree d, defined over a finite field F q ,  ... 
doi:10.1007/s10623-013-9911-7 fatcat:rzmmahrtyjcp5oz5wvwzposhbe

A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes [article]

Hang Chen, Cunsheng Ding, Sihem Mesnager, Chunming Tang
2020 arXiv   pre-print
The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity.  ...  More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented.  ...  Next we shall present several classes of minimal codes contained in the shortened second-order Reed-Muller code SRM(2, m). Theorem 28. Let m ≥ 3 be an odd integer.  ... 
arXiv:2004.04932v1 fatcat:cakvn4mttbflrm2edm4kobkgw4

Remarks on low weight codewords of generalized affine and projective Reed-Muller codes [article]

Stéphane Ballet, Robert Rolland
2013 arXiv   pre-print
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes.  ...  In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.  ...  Introduction -Notations This paper proposes an overview of the low weight codewords of generalized Reed-Muller codes and projective generalized Reed-Muller codes called respectively GRM codes and PGRM  ... 
arXiv:1203.4592v4 fatcat:acafz4hojnhu5gemqh4lacji24

The linear codes of t-designs held in the Reed-Muller and Simplex codes [article]

Cunsheng Ding, Chunming Tang
2020 arXiv   pre-print
On the other hand, some linear codes hold t-designs for some t ≥ 1. The purpose of this paper is to study the linear codes of some t-designs held in the Reed-Muller and Simplex codes.  ...  The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q).  ...  is equivalent to a concatenation of q − 1 copies the first-order projective Reed-Muller code.  ... 
arXiv:2008.09935v1 fatcat:fziupzzkubgftpn5wgxxvyqe4i

Symmetries of weight enumerators and applications to Reed-Muller codes [article]

Martino Borello, Olivier Mila
2018 arXiv   pre-print
to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.  ...  Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades.  ...  Moreover, the authors warmly thank the two anonymous reviewers which helped to improve a previous version of this paper.  ... 
arXiv:1707.00575v2 fatcat:vdjdrvbkmvgz7kxxr4ithpkpny
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