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Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Permutation Codes [article]

Michael Tait, Alexander Vardy, Jacques Verstraete
2013 arXiv   pre-print
Herein, we consider the situation where the ratio d/n is fixed and improve the Gilbert-Varshamov bound by a factor that is linear in n. That is, we show that if d/n < 0.5, then M(n,d)≥ cn n!  ...  Given positive integers n and d, let M(n,d) denote the maximum size of a permutation code of length n and minimum Hamming distance d. The Gilbert-Varshamov bound asserts that M(n,d) ≥ n!  ...  This technique has been introduced in [20] to improve the Gilbert-Varshamov bound on the size of binary codes.  ... 
arXiv:1311.4925v1 fatcat:mctfcoo7bjfuzhbgutyi4g7ng4

Asymptotic Improvement of the Gilbert–Varshamov Bound for Linear Codes

Philippe Gaborit, Gilles Zemor
2008 IEEE Transactions on Information Theory  
In this paper we show that certain asymptotic families of linear binary [n,n/2] random double circulant codes satisfy the same improved Gilbert-Varshamov bound.  ...  The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1) where V(n,d) stands for the volume of a Hamming  ...  In this paper we also improve on the the Gilbert-Varshamov bound by a linear factor in the block length but for linear codes, thereby solving one of the open problems of [6] .  ... 
doi:10.1109/tit.2008.928288 fatcat:o7nkkzlbazgvngkvfhvpo2rjiq

Improved code-based identification scheme [article]

Pierre-Louis Cayrel, Pascal Veron
2010 arXiv   pre-print
The proposed scheme is zero-knowledge and relies on an NP-complete problem coming from coding theory (namely the q-ary Syndrome Decoding problem).  ...  Taking into account a recent study of a generalization of Stern's information-set-decoding algorithm for decoding linear codes over arbitrary finite fields Fq we suggest parameters so that the public key  ...  ACKNOWLEDGMENT The authors want to thank Robert Niebuhr and Philippe Gaborit for their comments during the preparation of this paper and Christiane Peters for comments on her code.  ... 
arXiv:1001.3017v1 fatcat:hrhpjkih7nb3bl3jysdqefzlei

Improved constructions of nested code pairs [article]

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2017 arXiv   pre-print
The new constructions result from carefully applying the Feng-Rao bounds [18,27] to a family of codes defined from multivariate polynomials and Cartesian product point sets.  ...  By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large.  ...  Acknowledgments The authors thank Ryutaroh Matsumoto for pleasant discussions and the anony-  ... 
arXiv:1610.06363v3 fatcat:pnlsoiwrlncdldzen3scubzyhy

Improved Constructions of Nested Code Pairs

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2018 IEEE Transactions on Information Theory  
The new constructions result from carefully applying the Feng-Rao bounds [21, 31] to a family of codes defined from multivariate polynomials and Cartesian product point sets.  ...  By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large.  ...  Acknowledgments The authors thank Ryutaroh Matsumoto for pleasant discussions and the anony-  ... 
doi:10.1109/tit.2017.2755682 fatcat:xlf45hnajrbhblqnidv57p4thm

Improved Coding over Sets for DNA-Based Data Storage [article]

Hengjia Wei, Moshe Schwartz
2021 arXiv   pre-print
Various parameter regimes are studied. New bounds on code parameters are provided, which improve upon known bounds.  ...  New codes are constructed, at times matching the bounds up to lower-or der terms or small constant factors.  ...  The Gilbert-Varshamov bound shows that the redundancy of optimal (0, t, ε) D -correcting codes is at most t log M + 2tε log(L/2), see [10, Thm. 4] .  ... 
arXiv:2009.08816v2 fatcat:4xsgvbq4ebgvtgoii6riadjifq

Improvement on Parameters of Algebraic-Geometry Codes from Hermitian Curves [article]

Siman Yang
2007 arXiv   pre-print
This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in [9,10].  ...  Goppa's construction of Algebraic-Geometry codes leads to the Tsfasman-Vladut-Zink bound [1] which is a breakthrough in coding theory as it beats the Gilbert-Varshamov bound in an open interval over  ...  For fixed positive integers s > m define N s,m = |{ P ∈I P + D : I ⊆ {P 1 , . . . , P n }, |I| = m, D ≥ 0, D = sm}|. (2.1) Suppose N s,m < h(F ) (denoted by h(F ) the class number of F ).  ... 
arXiv:0709.1983v1 fatcat:tycfpqsbcrdxlhuoswxxfik3zy

Improved Constructions for Non-adaptive Threshold Group Testing [article]

Mahdi Cheraghchi
2013 arXiv   pre-print
The number of measurements resulting from this scheme is ideally bounded by O(d^g+3 ( d) n).  ...  This significantly improves the previously known (non-constructive) upper bound O(d^u+1(n/d)).  ...  on the the Tsfasman-Vlȃduţ-Zink (TVZ) bound, Hermitian codes, and finally, codes on the Gilbert-Varshamov (GV) bound.  ... 
arXiv:1002.2244v3 fatcat:sitnmasbeja5va6mg3anztuubu

Further improvements on asymptotic bounds for codes using distinguished divisors

Harald Niederreiter, Ferruh Özbudak
2007 Finite Fields and Their Applications  
In recent years the Tsfasman-Vlȃduţ-Zink lower bound on α q (δ) was improved by Elkies, Xing, and Niederreiter and Özbudak.  ...  In this paper we show further improvements on these bounds by using distinguished divisors of global function fields.  ...  Acknowledgments The second author is partially supported by the Turkish Academy of Sciences in the framework of the Young Scientists Award Programme (F.Ö./TÜBA-GEBIP/2003-13).  ... 
doi:10.1016/j.ffa.2005.11.004 fatcat:3kj65j2vdrfj5a74w3cjkv3k2e

Asymptotic Gilbert-Varshamov bound on Frequency Hopping Sequences [article]

Xianhua Niu, Chaoping Xing, Chen Yuan
2018 arXiv   pre-print
In particular, the most important lower bound--the Gilbert-Varshamov bound in coding theory has not been transformed to frequency hopping sequence sets.  ...  We provide two proofs of the Gilbert-Varshamov bound. One is based on probabilistic method that requires advanced tool--martingale. This proof covers the whole rate region.  ...  Our lower bound comes from the classic Gilbert-Varshamov bound. However, the original Gilbert-Varshamov bound does not hold for cyclic codes.  ... 
arXiv:1810.11757v2 fatcat:e6a7wvplrff5rjz7zpagsvejwe

Strengthening the Gilbert–Varshamov bound

Alexander Barg, Sugi Guritman, Juriaan Simonis
2000 Linear Algebra and its Applications  
The paper discusses some ways to strengthen (nonasymptotically) the Gilbert-Varshamov bound for linear codes.  ...  The unifying idea is to study a certain graph constructed on vectors of low weight in the cosets of the code, which we call the Varshamov graph.  ...  To improve the Varshamov-Gilbert bound asymptotically is a notoriously difficult task [11] .  ... 
doi:10.1016/s0024-3795(99)00271-2 fatcat:oml44tl7inbsdladwel5xbruba

Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data [article]

Gautam Kamath, Xingtu Liu, Huanyu Zhang
2021 arXiv   pre-print
We provide improved upper bounds on the excess population risk under concentrated DP for convex and strongly convex loss functions.  ...  Instead, as introduced by Wang, Xiao, Devadas, and Xu , we study general convex loss functions with the assumption that the distribution of gradients has bounded k-th moments.  ...  Note that the distribution of X is a mixture of M distributions. Specifically, for any event S, Pr (X ∈ S) = 1 M i∈M Pr X∼p n i (X ∈ S).  ... 
arXiv:2106.01336v4 fatcat:qer7h2y7c5ctjmthuvdcnhz5bm

Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric [article]

Moshe Schwartz, Pascal O. Vontobel
2017 arXiv   pre-print
Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.  ...  These new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0.029 bits per symbol.  ...  It is an important component in many bounds on code parameters, most notably, the ballpacking bound and the Gilbert-Varshamov bound [33] .  ... 
arXiv:1609.05277v2 fatcat:o6gowpoazbb4di6b56npczokva

Construction of asymptotically good locally repairable codes via automorphism groups of function fields [article]

Xudong Li, Liming Ma, Chaoping Xing
2017 arXiv   pre-print
Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size q.  ...  In this paper, we extend the construction given in BTV17 via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality.  ...  An easy computation shows that d m + r + 1 r k m n m − (r + 1)s − (r − 1) m−1 + (r + 1)s − (g(T m ) − 1) n m − (r − 1) m−1 − g(T m ) + 1 n m − (r − 1) m−1 − m from Theorem 3.3 and Proposition 2.1(iv).  ... 
arXiv:1711.07703v1 fatcat:mp3q6mrvmvecvdsw2gugfz3pou

On Weierstrass semigroups and the redundancy of improved geometric Goppa codes

R. Pellikaan, F. Torres
1999 IEEE Transactions on Information Theory  
Improved geometric Goppa codes have a smaller redundancy and the same bound on the minimum as ordinary algebraic geometry codes.  ...  For an asymptotically good sequence of function fields we give a formula for the redundancy.  ...  Let T m be obtained from T m−1 by adjoining a new element x m that satisfies the equation: x r m + x m = x r m−1 x r−1 m−1 + 1 .  ... 
doi:10.1109/18.796393 fatcat:4wkwcsbf3bcl3humfre5c2ykd4
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