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Relative Generalized Hamming weights of affine Cartesian codes [article]

Mrinmoy Datta
2019 arXiv   pre-print
of the affine Cartesian codes by Beelen and Datta.  ...  We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics.  ...  Several articles, for example [3, 4] , are devoted towards the determination of the next to minimal weights of affine Cartesian codes.  ... 
arXiv:1909.06138v1 fatcat:363uyrwdijdkhpswc2x725tr34

Relative generalized Hamming weights of affine Cartesian codes

Mrinmoy Datta
2020 Designs, Codes and Cryptography  
of the affine Cartesian codes by Beelen and Datta.  ...  We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics.  ...  Several articles, for example [3, 4] , are devoted towards the determination of the next to minimal weights of affine Cartesian codes.  ... 
doi:10.1007/s10623-020-00745-8 fatcat:hyh35ycnn5dsplfhbo3wh4quoq

Relative generalized Hamming weights of evaluation codes [article]

Delio Jaramillo-Velez and Hiram H. López and Yuriko Pitones
2021 arXiv   pre-print
We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.  ...  The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound.  ...  In Section 4, we present the next-to-minimal weights of toric codes over hypersimplices of degree one.  ... 
arXiv:2112.07085v1 fatcat:3i7mpvs6cbgr5crl3f6mq7jkra

On the second Hamming weight of some Reed-Muller type codes [article]

Cicero Carvalho
2013 arXiv   pre-print
Using methods from Gr\"obner basis theory we determine the second Hamming weight (also called next-to-minimal weight) for particular cases of affine cartesian codes and also some higher Hamming weights  ...  We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q.  ...  In the next section we will determine the exact value of the second Hamming weight, also called next-to-minimal weight, for some particular cases of C(d) as well as some higher Hamming weights of these  ... 
arXiv:1306.4727v2 fatcat:3kxdb74msjd7loxxm3yiak55hm

Batch Codes from Affine Cartesian Codes and Quotient Spaces [article]

Travis Baumbaugh, Haley Colgate, Timothy Jackman, Felice Manganiello
2020 arXiv   pre-print
We are able to prove that under these conditions, an affine Cartesian code is able to satisfy a query of size up to one more than the dimension of the space of the ambient space.  ...  Affine Cartesian codes are defined by evaluating multivariate polynomials at a cartesian product of finite subsets of a finite field. In this work we examine properties of these codes as batch codes.  ...  We already recalled that affine Cartesian codes are a generalization of Reed-Muller codes. For simplicity of notation, the remainder of this section and the next section focus on Reed-Muller codes.  ... 
arXiv:2005.07577v1 fatcat:avy3v3lb4zdzhit5uhnclau4yy

Affine Cartesian codes with complementary duals [article]

Hiram H. López, Felice Manganiello, Gretchen L. Matthews
2018 arXiv   pre-print
Generalized affine Cartesian codes arise naturally as the duals of affine Cartesian codes in the same way that generalized Reed-Solomon codes arise as duals of Reed-Solomon codes.  ...  A linear code C with the property that C ∩ C^ = {0 } is said to be a linear complementary dual, or LCD, code. In this paper, we consider generalized affine Cartesian codes which are LCD.  ...  In [7] the authors find several values for the second least weight of codewords, also known as the next-to-minimal Hamming weight.  ... 
arXiv:1805.07018v2 fatcat:df6v7g2wknejfngzhb6wuluzae

An extension of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords to a class of Reed-Muller type codes [article]

Cicero Carvalho, Victor G.L. Neumann
2019 arXiv   pre-print
The paper also brings an expository section on the study of the structure of low weight codewords, not only for affine Reed-Muller type codes, but also for the projective ones.  ...  In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed-Muller codes where, among many important results, they proved that the minimal weight codewords of these codes  ...  The next result describes the minimal weight codewords of affine cartesian codes for the lowest range of values of d, meaning the case when k = 0.  ... 
arXiv:1903.09458v1 fatcat:32klqszghbhi7msbtsag3rbvcu

Shape-preserving, multiscale interpolation by univariate curvature-based cubic L1 splines in Cartesian and polar coordinates

John E. Lavery
2002 Computer Aided Geometric Design  
The coefficients of these splines are calculated by minimizing the L 1 norm of curvature.  ...  In computational experiments in Cartesian coordinates, cubic L 1 splines based on curvature preserve the shape of multiscale data well, as do cubic L 1 splines based on the second derivative.  ...  system based on the Newton equations and the weights for the next least-squares system are calculated using the residuals of the original overdetermined system.  ... 
doi:10.1016/s0167-8396(02)00087-0 fatcat:wa3cywlmjzhplacgokexbt5lou

Quantum codes from affine variety codes and their subfield-subcodes [article]

Carlos Galindo, Fernando Hernando
2014 arXiv   pre-print
We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction.  ...  With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.  ...  The weight function allows us to define the so-called Feng-Rao bound on the minimum distance of the previous codes.  ... 
arXiv:1403.4060v2 fatcat:koyl7hargzcclkuustlb663vqy

A family of codes with locality containing optimal codes [article]

Bruno Andrade, Cícero Carvalho, Victor G.L. Neumann, Antônio C.P. Veiga
2021 arXiv   pre-print
We determine the dimension of these codes, and also bounds for the minimum distance.  ...  Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes.  ...  cartesian codes to the minimum distance of affine cartesian codes.  ... 
arXiv:2101.07629v1 fatcat:zvmjsns3urcb7jlkqqtbwlhp3m

Quantum codes from affine variety codes and their subfield-subcodes

Carlos Galindo, Fernando Hernando
2014 Designs, Codes and Cryptography  
We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction.  ...  With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.  ...  Next, we will show a table containing parameters corresponding with some quantum codes coming from subfield-subcodes of affine variety codes.  ... 
doi:10.1007/s10623-014-0016-8 fatcat:lkpxzvnfqbhsfproy5hdafnsbu

Generalized minimum distance functions [article]

Manuel Gonzalez-Sarabia, Jose Martínez-Bernal, Rafael H. Villarreal, Carlos E. Vivares
2018 arXiv   pre-print
Then we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code.  ...  We show that the generalized footprint function of I is a lower bound for the r-th generalized Hamming weight of C_X(d). Then we present some applications to projective nested cartesian codes.  ...  It is an interesting problem to find alternative, easy to evaluate formulas for the r-th generalized Hamming weight of an affine cartesian code.  ... 
arXiv:1707.03285v4 fatcat:n5gcighy35evrogdrsnckvpazm

Smooth Surfaces via Nets of Geodesics [article]

Tom Gilat
2022 arXiv   pre-print
It is based on a theoretical result by the author regarding minimal Gaussian curvature surfaces with geodesic boundary conditions.  ...  The novelty of the method is that it consists of the computation of each patch in the net independently with the union of the patches being a smooth surface.  ...  We need to heuristically fix an affine plane for which the projection of the minimal Gaussian curvature surface spanning Γ would be one-to-one.  ... 
arXiv:2109.01429v2 fatcat:lawgxo4rurba7kobadk2xknczm

Generalized minimum distance functions

Manuel González-Sarabia, José Martínez-Bernal, Rafael H. Villarreal, Carlos E. Vivares
2018 Journal of Algebraic Combinatorics  
Then, we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine Cartesian code.  ...  We show that the generalized footprint function of I is a lower bound for the r th generalized Hamming weight of C X (d). Then, we present some applications to projective nested Cartesian codes.  ...  Acknowledgements We thank the referees for a careful reading of the paper and for the improvements suggested.  ... 
doi:10.1007/s10801-018-0855-x fatcat:4nenvmcp2zhgriguctmfvzw4zi

Efficient maximum-likelihood decoding of spherical lattice codes

Karen Su, Inaki Berenguer, Ian Wassell, Xiaodong Wang
2009 IEEE Transactions on Communications  
Recently, spherical Lattice Space-Time (LAST) codes were proposed to realize the optimal diversitymultiplexing tradeoff of MIMO channels.  ...  Lattice codes have long been of interest due to their rich structure, leading to numerous decoding algorithms for unbounded lattices, as well as those with axis-aligned rectangular shaping regions.  ...  It is advantageous to consider the minimization problem from the perspective of (6) because then the search set has an underlying Cartesian product structure Z m that lends itself easily to divide and  ... 
doi:10.1109/tcomm.2009.08.070329 fatcat:glwsh4jlrramlma5meyhepahhq
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