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Batch Codes from Hamming and Reed-Müller Codes [article]

Travis Baumbaugh, Yariana Diaz, Sophia Friesenhahn, Felice Manganiello, Alexander Vetter
2017 arXiv   pre-print
We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Müller codes.  ...  We then show that binary first order Reed-Müller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Müller codes which have order less than  ...  The REU was made possible by an NSF Research Training Group (RTG) grant (DMS #1547399) promoting Coding Theory, Cryptography, and Number Theory at Clemson.  ... 
arXiv:1710.07386v1 fatcat:l74bhyftonefdnlamtetk6v2ry

Batch codes from Hamming and Reed-Muller codes

Travis Baumbaugh, Yariana Diaz, Sophia Friesenhahn, Felice Manganiello, Alexander Vetter
2018 Journal of Algebra Combinatorics Discrete Structures and Applications  
We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Muller codes.  ...  We then show that binary first order Reed-Muller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Muller codes which have order less than  ...  We conclude the section by introducing a concept of optimal batch codes. After the background, we study the batch properties of binary Hamming codes and Reed-Muller codes.  ... 
doi:10.13069/jacodesmath.466634 fatcat:smqutqrjivhp3kkfjx2fwmovj4

Binary, Shortened Projective Reed Muller Codes for Coded Private Information Retrieval [article]

Myna Vajha, Vinayak Ramkumar, P. Vijay Kumar
2017 arXiv   pre-print
These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary, Projective Reed Muller (PRM) code.  ...  It follows from a result by Vardy and Yaakobi, that these codes also yield optimal, systematic primitive multi-set (n, k, τ)_B batch codes for τ=3,4.  ...  The restriction of evaluation to homogeneous polynomials takes us from the setting of conventional and affine RM codes to the setting of Projective Reed-Muller (PRM) codes.  ... 
arXiv:1702.05074v2 fatcat:g5ex7cxahrf4bm236yrgnxen6a

Page 4870 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
(F-LIMO-OM; Limoges) ; de Maximy, Louis (F-LIMO-OM; Limoges) Cyclic projective Reed-Muller codes.  ...  Summary: “The projective Reed-Muller codes (PRM codes) were introduced by G. Lachaud [in Coding theory and applications (Cachan, 1986), 125-129, Lecture Notes in Comput.  ... 

Service Rate Region: A New Aspect of Coded Distributed System Design [article]

Mehmet Aktas, Gauri Joshi, Swanand Kadhe, Fatemeh Kazemi, Emina Soljanin
2021 arXiv   pre-print
We also discover fundamental connections between multi-set batch codes and the problem of maximizing the service rate region.  ...  We show the effectiveness of hybrid codes that combine replication and erasure coding in terms of code design.  ...  ) codes, and first-order Reed-Muller (RM) codes.  ... 
arXiv:2009.01598v2 fatcat:6kpn43dkmfcpdhmcj2qovkzzf4

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

Travis Baumbaugh, Haley Colgate, Timothy Jackman, Felice Manganiello
2020 arXiv   pre-print
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 consider the recovery sets to be defined by points aligned on a specific direction and the buckets to be derived from cosets of a subspace of the ambient space of the evaluation points.  ...  This is a generalization of work in [6] , and in a similar fashion, the work in this paper aims to expand the study of batch properties from Reed-Muller codes as studied in [1] to the broader class  ... 
arXiv:2005.07577v1 fatcat:avy3v3lb4zdzhit5uhnclau4yy

Algebraic Coding Theory for Networks, Storage, and Security (Dagstuhl Seminar 18511)

Eimear Byrne, Martin Bossert, Antonia Wachter-Zeh, Michael Wagner
2019 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 18511 "Algebraic Coding Theory for Networks, Storage, and Security".  ...  The ever increasing traffic in networks and the growth of distributed storage systems require advanced techniques based on algebraic coding to meet user demand.  ...  Consider bivariate Gabidulin "Reed-Muller" codes over general finite fields, as discussed in the working group on algebraic geometry codes.  ... 
doi:10.4230/dagrep.8.12.49 dblp:journals/dagstuhl-reports/ByrneBW18 fatcat:qmcdjxigx5ahxe3jpnmns4v6ka

Batch and PIR Codes and Their Connections to Locally Repairable Codes [article]

Vitaly Skachek
2017 arXiv   pre-print
In this survey, two related families of codes are discussed: batch codes and codes for private information retrieval.  ...  Bounds on the parameters of the codes, as well as basic constructions, are presented. Connections between different code families are discussed.  ...  This work is supported in part by the grants PUT405 and IUT2-1 from the Estonian Research Council and by the EU COST Action IC1104.  ... 
arXiv:1611.09914v4 fatcat:qmj3i24exjcnjmuykep2h3aowq

Hardness of Approximate Nearest Neighbor Search [article]

Aviad Rubinstein
2018 arXiv   pre-print
Efficient PCPs from AG codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but our construction is the first to yield new hardness results.  ...  We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance.  ...  (AG) codes generalize Reed Solomon and Reed Muller codes.  ... 
arXiv:1803.00904v1 fatcat:7bzhtsporbadvakg5a27uh445m

KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning [article]

Ashok Vardhan Makkuva, Xiyang Liu, Mohammad Vahid Jamali, Hessam Mahdavifar, Sewoong Oh, Pramod Viswanath
2021 arXiv   pre-print
Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes: each is a linear code and represents a mathematical breakthrough.  ...  KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding, in the challenging short-to-medium block length regime on the AWGN channel.  ...  Acknowledgements Ashok would like to thank his colleagues Mona Zehni and Konik Kothari for helpful discussions about the project.  ... 
arXiv:2108.12920v1 fatcat:rmv2iefpfjacxctqscacxvxaza

Reed-Muller Subcodes: Machine Learning-Aided Design of Efficient Soft Recursive Decoding [article]

Mohammad Vahid Jamali, Xiyang Liu, Ashok Vardhan Makkuva, Hessam Mahdavifar, Sewoong Oh, Pramod Viswanath
2021 arXiv   pre-print
Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms  ...  In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to n, where n is the blocklength.  ...  INTRODUCTION Reed-Muller (RM) codes are among the oldest families of error-correcting codes, and their origin backs to almost seven decades ago [1] , [2] .  ... 
arXiv:2102.01671v1 fatcat:7ycflgbs4feahgfn6g5hlypyya

Dual-Code Bounds on Multiple Concurrent (Local) Data Recovery [article]

Gianira N. Alfarano, Alberto Ravagnani, Emina Soljanin
2022 arXiv   pre-print
This paper initiates a study of such systems within the classical coding theory.  ...  Remark 4 . 8 . 48 The previous corollary on MDS codes is sharp whenever k ≤ n − k; see [7, Theorem 2]. Example 4 . 12 ( 412 First-Order Reed-Muller Code).  ...  Consider the following non-systematic generator matrix G ∈ F 4×8 2 of the first order Reed-Muller code, namely 1 1 = δ 1 2 = δ 2 2 = δ 2 2 = 5 , δ 1 4 = 2 , ω 4 = 1 and δ 2 4 = 4 . 222542444 = δ 1 3  ... 
arXiv:2201.07503v2 fatcat:63tvbehcwjaerfglbw5kb3yvum

Lengthening and Extending Binary Private Information Retrieval Codes [article]

Hsuan-Yin Lin, Eirik Rosnes
2018 arXiv   pre-print
symbol and code extension by at most t/2 code symbols.  ...  In this work, we show that a family of t-server PIR codes (with increasing dimensions and blocklengths) can be constructed from an existing t-server PIR code through lengthening by a single information  ...  In the recent work of [6] , the authors found that the so-called shortened projective Reed Muller (SPRM) codes are good t-server PIR codes for t = 2 ℓ − 1 and 2 ℓ where ℓ is a positive integer.  ... 
arXiv:1707.03495v3 fatcat:kjd6ly2z2rarlaspeczabk4kvq

Post-Quantum and Code-Based Cryptography—Some Prospective Research Directions

Chithralekha Balamurugan, Kalpana Singh, Ganeshvani Ganesan, Muttukrishnan Rajarajan
2021 Cryptography  
Some potential research directions that are yet to be explored in code-based cryptography research from the perspective of codes is a key contribution of this paper.  ...  This paper provides an overview of the various research directions that have been explored in post-quantum cryptography and, specifically, the various code-based cryptography research dimensions that have  ...  Acknowledgments: This research work has been carried out under the leadership of the Institute for Technological Research SystemX, and therefore granted within the scope of the program "Recherche Exploratoire  ... 
doi:10.3390/cryptography5040038 fatcat:wkb43zunwjagbnljc6brfou4ty

Recent Progress in Code-Based Cryptography [chapter]

Pierre-Louis Cayrel, Sidi Mohamed El Yousfi Alaoui, Gerhard Hoffmann, Mohammed Meziani, Robert Niebuhr
2011 Communications in Computer and Information Science  
Section 3 deals with the new improvements of code-based signature schemes and Section 4 presents the new results in code-based secret-key cryptography.  ...  The last three years have witnessed tremendous progress in the understanding of code-based cryptography.  ...  Reed-Muller codes using Stern's algorithm [75] .  ... 
doi:10.1007/978-3-642-23141-4_3 fatcat:tydg6d5zunhbplpchcnozoaanm
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