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Lifted Projective Reed-Solomon Codes [article]

Julien Lavauzelle
2018 arXiv   pre-print
We also prove that the lifting of both Reed-Solomon and projective Reed-Solomon codes are deeply linked through shortening and puncturing operations.  ...  We finally emphasize the practicality of lifted projective Reed-Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms.  ...  Projective Reed-Solomon codes are also called extended, or doubly-extended Reed-Solomon codes.  ... 
arXiv:1809.00931v1 fatcat:brcnahpejjf2josfm3vngkcpn4

Lifted projective Reed–Solomon codes

Julien Lavauzelle
2018 Designs, Codes and Cryptography  
We also prove that the lifting of both Reed-Solomon and projective Reed-Solomon codes are deeply linked through shortening and puncturing operations.  ...  Resulting codes are named lifted Reed-Solomon codes, and in this work, more shortly referred to as affine lifted codes. Organisation.  ...  Projective Reed-Solomon codes are also called extended, or doubly-extended Reed-Solomon codes.  ... 
doi:10.1007/s10623-018-0552-8 fatcat:swq7fs34ebcctaqhroxq7zvlt4

Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information Retrieval [article]

Julien Lavauzelle, Jade Nardi
2019 arXiv   pre-print
This lead the community to design codes with similar local properties but larger dimension, namely the lifted Reed-Solomon codes.  ...  With a careful analysis of their degree sets, we notably show that the weighted lifting of Reed-Solomon codes produces families of codes with remarkable asymptotic parameters.  ...  Similarly to the constructions of lifted (affine [GKS13] and projective [Lav18b] ) Reed-Solomon codes and lifted Hermitian codes [Guo16] , we also prove that for fixed η and q → ∞, weighted lifts of  ... 
arXiv:1904.08696v1 fatcat:rv33e32ijfadhl266nn2t6xrf4

Lifted Reed-Solomon Codes with Application to Batch Codes [article]

Lukas Holzbaur and Rina Polyanskaya and Nikita Polyanskii and Ilya Vorobyev
2020 arXiv   pre-print
Lifted Reed-Solomon (RS) codes are defined as the evaluation of polynomials in a vector space over a field by requiring their restriction to every line in the space to be a codeword of the RS code.  ...  First, we improve the estimate of the code rate of lifted RS codes for lifting parameter m> 3 and large field size.  ...  Lifted Reed-Solomon codes Let us recall the definition of lifted Reed-Solomon codes introduced in [12] in a more general form. Definition 1 (Lifted Reed-Solomon code, [12] ).  ... 
arXiv:2001.11981v2 fatcat:fgvxlkcwkjbxrizmtiht4t4yci

Twisted Reed-Solomon Codes With One-dimensional Hull [article]

Yansheng Wu
2020 arXiv   pre-print
In this short letter, we obtain some twisted Reed-Solomon MDS codes with one-dimensional hull. Moreover, these codes are not monomially equivalent to Reed-Solomon codes.  ...  Maximum distance separable (MDS) codes are codes meeting the Singleton bound. Twisted Reed-Solomon codes is a generalization of Reed-Solomon codes, which is also a nice construction for MDS codes.  ...  An effective method for construction of twisted Reed-Solomon codes with MDS property is to use the lifting of the finite field (refer to [2] ).  ... 
arXiv:2009.07743v1 fatcat:oxaccaf5yzcfzljhdh3kaah2uq

Quadratic-Curve-Lifted Reed-Solomon Codes [article]

Hedongliang Liu, Lukas Holzbaur, Nikita Polyanskii, Sven Puchinger, Antonia Wachter-Zeh
2022 arXiv   pre-print
In this work we introduce and study quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, where the codeword symbols whose coordinates are on a quadratic curve form a codeword of a Reed-Solomon code.  ...  Moreover, we provide analytical results on the minimum distance of this class of codes and compare QC-LRS codes with lifted Reed-Solomon codes by simulations in terms of the local recovery capability against  ...  Then 3 Quadratic-Curve-Lifted Reed-Solomon Codes a b = ℓ i=1 a i b i mod p .  ... 
arXiv:2109.14478v3 fatcat:5tfk7y5ibzfbdovvhl3sgghjh4

Applications of list decoding to tracing traitors

A. Silverberg, J. Staddon, J.L. Walker
2003 IEEE Transactions on Information Theory  
Index Terms-Algebraic geometry code, identifiable parent property, list decoding, traceability code, traitor tracing, Reed-Solomon code. A. Silverberg is with the  ...  We apply results from algebraic coding theory to solve problems in cryptography, by using recent results on list decoding of error-correcting codes to efficiently find traitors who collude to create pirates  ...  Reed-Solomon codes can be viewed as algebraic geometry codes by taking X to be the projective line, P to be the set of points corresponding to the r chosen field elements, P 0 to be the point at infinity  ... 
doi:10.1109/tit.2003.810630 fatcat:p7zwver565hifban5ey7e247vi

Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces [article]

Matilde Marcolli
2018 arXiv   pre-print
It discusses some general approaches to the construction of classical and quantum holographic codes on Bruhat-Tits trees and buildings and on Drinfeld symmetric spaces, in the context of the p-adic AdS  ...  Reed-Solomon Codes and Quantum Algebro-Geometric Codes.  ...  2q + 1, q + 1]] q -quantum Reed-Solomon codê C.  ... 
arXiv:1801.09623v1 fatcat:csfumw664zhljaxyjv3ylkllse

Efficient Traitor Tracing Algorithms Using List Decoding [chapter]

Alice Silverberg, Jessica Staddon, Judy L. Walker
2001 Lecture Notes in Computer Science  
We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors.  ...  Reed-Solomon or AG code and the "inner code" is a Hadamard code).  ...  Reed-Solomon codes can be viewed as algebraic geometry codes by taking X to be the projective line, P to be the set of points corresponding to the r chosen field elements, P 0 to be the point at infinity  ... 
doi:10.1007/3-540-45682-1_11 fatcat:nepsczcfpzctdd57r4stwlndpm

The geometry of one-weight codes in the sum-rank metric [article]

Alessandro Neri and Paolo Santonastaso and Ferdinando Zullo
2021 arXiv   pre-print
This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them.  ...  code.  ...  The research of the last author was supported by the project "VALERE: VanviteLli pEr la RicErca" of the University of Campania "Luigi Vanvitelli".  ... 
arXiv:2112.04989v1 fatcat:lrhrfcqavfaqfayedpt6jzybma

Lifted Reed-Solomon Codes and Lifted Multiplicity Codes [article]

Lukas Holzbaur, Rina Polyanskaya, Nikita Polyanskii, Ilya Vorobyev, Eitan Yaakobi
2021 arXiv   pre-print
Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed from specific sets of m-variate polynomials.  ...  Using these results, we give a new explicit construction of batch codes utilizing lifted Reed-Solomon codes.  ...  Lifted Reed-Solomon codes Let us recall the definition of lifted Reed-Solomon codes introduced in [16] . Definition 1 (Lifted Reed-Solomon code, [16] ).  ... 
arXiv:2110.02008v2 fatcat:25ctwd2qt5ainpfqmjhw2ecrgq

Construction of Adaptive Short LDPC Codes for Distributed Transmit Beamforming [article]

Ismail Shakeel, Ishtiaq Ahmad, Hajime Suzuki
2018 arXiv   pre-print
This paper proposes an adaptive Low Density Parity Check (LDPC) coding scheme for the DTB system. The proposed scheme constructs powerful LDPC codes with varying code-rates and block-lengths.  ...  The performance of some of the codes constructed using the proposed scheme is evaluated and compared with the uncoded and other coded-DTB systems.  ...  The authors would like to acknowledge Dr Gerald Bolding (DST Group) for the valuable suggestions made to improve the P-LDPC code construction scheme described in this paper.  ... 
arXiv:1807.03526v2 fatcat:qfyybevsnjbxddsrha5hdaolca

Automorphism groups of generalized Reed-Solomon codes [article]

David Joyner, Amy Ksir, Will Traves
2008 arXiv   pre-print
We also give examples of generalized Reed-Solomon codes with large automorphism groups G, such as G=PSL(2,q), and explicitly describe their G-module structure.  ...  We (re)classify those finite groups that can arise as the automorphism group of an AG code for the projective line and give an explicit description of how these groups appear.  ...  Introduction Reed-Solomon codes are popular in applications because fast encoding and decoding algorithms are known for them.  ... 
arXiv:0801.4007v1 fatcat:f7bmn7ligna23jmtz35veb35pe

Orbit codes — A new concept in the area of network coding

Anna-Lena Trautmann, Felice Manganiello, Joachim Rosenthal
2010 2010 IEEE Information Theory Workshop  
We introduce a new class of constant dimension codes called orbit codes. The basic properties of these codes are derived.  ...  It will be shown that many of the known families of constant dimension codes in the literature are actually orbit codes.  ...  In [1] a Reed-Solomon-like construction for constant dimension codes was introduced and Silva et al. showed that lifting (i.e. concatenating an identity block in front of a matrix and taking the row  ... 
doi:10.1109/cig.2010.5592788 dblp:conf/itw/TrautmannMR10 fatcat:mb5khgecuncajomrjpatkogeeu

Page 7009 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews  
Summary: “Inspired by Sudan’s recent algorithm for Reed- Solomon codes [M. Sudan, J.  ...  The structure of the quaternary Preparata codes, the Z,-linear lift of the binary Golay codes and the Zo-linear lift of the ternary Golay codes are studied in Chapter 4.  ... 
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