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Relative Generalized Hamming weights of affine Cartesian codes
[article]
2019
arXiv
pre-print
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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
2020
Designs, Codes and Cryptography
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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
On the second Hamming weight of some Reed-Muller type codes
[article]
2013
arXiv
pre-print
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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
An extension of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords to a class of Reed-Muller type codes
[article]
2019
arXiv
pre-print
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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
Affine Cartesian codes with complementary duals
[article]
2018
arXiv
pre-print
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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
Relative generalized Hamming weights of evaluation codes
[article]
2021
arXiv
pre-print
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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. ...
More recently, Carvalho and Neumann have determined the next-to-minimal weights of affine Cartesian codes, binary projective Reed-Muller codes, and projective Reed-Muller codes [5, 6, 7, 8, 9, 10] . ...
arXiv:2112.07085v1
fatcat:3i7mpvs6cbgr5crl3f6mq7jkra
Full Characterization of Minimal Linear Codes as Cutting Blocking Sets
[article]
2020
arXiv
pre-print
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In addition to this, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented. ...
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal ...
Some new infinite classes of minimal q-ary linear codes with w min w max ≤ q−1 q are derived. Finally we determine the weight distributions of two subfamilies of the proposed minimal codes. ...
arXiv:1911.09867v2
fatcat:qitb36vpgfeifktyc5wr4cwyze
Efficient maximum-likelihood decoding of spherical lattice codes
2009
IEEE Transactions on Communications
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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
Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement
[article]
2015
arXiv
pre-print
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of a new family of linear codes, the J-affine variety codes. ...
These codes are constructed with a new generalization of the Steane's enlargement procedure and by considering orthogonal subfield-subcodes --with respect to the Euclidean and Hermitian inner product-- ...
Acknowledgment The authors wish to thank Ryutaroh Matsumoto and the anonymous reviewers for helpful comments on this paper. ...
arXiv:1503.00879v2
fatcat:q2u4c4rpvzgvrao7ufivnvtgdy
Generalized minimum distance functions
[article]
2018
arXiv
pre-print
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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
Generalized minimum distance functions
2018
Journal of Algebraic Combinatorics
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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 space-time codes
2006
2006 IEEE International Conference on Communications
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Specifically we apply it to the spherical Lattice Space-Time (LAST) codes recently put forward by El Gamal et al. that have been proven to achieve the optimal diversity-multiplexing tradeoff of MIMO channels ...
We demonstrate its performance and complexity by applying two of the most efficient tree-based ML detectors currently reported in the literature to the spherical LAST code proposed for the 2 × 2 MIMO channel ...
Determining the candidate range We determine the candidate range by applying a sort of relaxation to the representation of the search set. ...
doi:10.1109/icc.2006.255265
dblp:conf/icc/SuBWW06
fatcat:wvgoxufggngmdhgksenrs52orq
Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement
2015
Quantum Information Processing
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a new family of linear codes, the J-affine variety codes. ...
These codes are constructed with a new generalization of the Steane's enlargement procedure and by considering orthogonal subfield-subcodes -with respect to the Euclidean and Hermitian inner product-of ...
Acknowledgment The authors wish to thank Ryutaroh Matsumoto for helpful comments on this paper. ...
doi:10.1007/s11128-015-1057-2
fatcat:b53llaif5fam5mktc2fk5wm6py
Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity
[article]
2020
arXiv
pre-print
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The methodology further links geometric variables of the input assembling units to a type of intuitive topological bar-code of the output atlas, which in turn determine stable assembled structures and ...
We use the novel convex Cayley (distance-based) parametrization that is unique to assembly, as opposed to folding. ...
The bar-code for the basin behaves similarly to the bar-code of the atlas. ...
arXiv:1203.3811v4
fatcat:whqecvwu5fauhmzfhtjw5wljpi
Automatic Correction of Loop Transformations
2007
Parallel Architecture and Compilation Techniques (PACT), Proceedings of the International Conference on
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We propose a general method to correct any sequence of loop transformations through a combination of loop shifting, code motion and index-set splitting. ...
We propose a new design where optimization heuristics first address the main performance anomalies, then correct potentially illegal loop transformations a posteriori, attempting to minimize the performance ...
We consider a dependence from S to T in the original code and we want to determine if it has been preserved in the transformed program. ...
doi:10.1109/pact.2007.4336220
fatcat:dsdif3ockjh57jhxuzky535vtu
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