Formalization of Error-Correcting Codes: From Hamming to Modern Coding Theory

2015 ◽  
pp. 17-33 ◽  
Author(s):  
Reynald Affeldt ◽  
Jacques Garrigue
Keyword(s):  
Coding Theory ◽  
Error Correcting Codes

scholarly journals A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

2019 ◽  
Vol 9 (2) ◽  
pp. 1232
Author(s):  
Issam Abderrahman Joundan ◽  
Said Nouh ◽  
Mohamed Azouazi ◽  
Abdelwahed Namir
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Search Problem ◽  
Public Key Cryptosystems ◽  
True Value ◽  
Efficient Scheme ◽  
Minimum Distances

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity.  ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

scholarly journals Hamming Codes: Error Reducing Techniques

2021 ◽  
Vol 9 (11) ◽  
pp. 1972-1974
Author(s):  
Rohitkumar R Upadhyay
Keyword(s):  
Lower Bound ◽  
Error Correction ◽  
Coding Theory ◽  
Hamming Distance ◽  
Error Reduction ◽  
Error Correcting Codes ◽  
Significant Other ◽  
Hamming Codes ◽  
Decoding Algorithms ◽  
Reducing Properties

Abstract: Hamming codes for all intents and purposes are the first nontrivial family of error-correcting codes that can actually correct one error in a block of binary symbols, which literally is fairly significant. In this paper we definitely extend the notion of error correction to error-reduction and particularly present particularly several decoding methods with the particularly goal of improving the error-reducing capabilities of Hamming codes, which is quite significant. First, the error-reducing properties of Hamming codes with pretty standard decoding definitely are demonstrated and explored. We show a sort of lower bound on the definitely average number of errors present in a decoded message when two errors for the most part are introduced by the channel for for all intents and purposes general Hamming codes, which actually is quite significant. Other decoding algorithms are investigated experimentally, and it generally is definitely found that these algorithms for the most part improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of for all intents and purposes standard decoding. Keywords: coding theory, hamming codes, hamming distance

scholarly journals GCD of Aunu Binary Polynomials of Cardinality Seven Using Extended Euclidean Algorithm

2021 ◽  
Vol 09 (09) ◽  
Author(s):  
Ibrahim A. A. ◽  
Keyword(s):  
Finite Fields ◽  
Coding Theory ◽  
Error Correcting Codes ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Algebraic Structures ◽  
Extended Euclidean Algorithm ◽  
Encryption Schemes ◽  
Theory Error

Finite fields is considered to be the most widely used algebraic structures today due to its applications in cryptography, coding theory, error correcting codes among others. This paper reports the use of extended Euclidean algorithm in computing the greatest common divisor (gcd) of Aunu binary polynomials of cardinality seven. Each class of the polynomial is permuted into pairs until all the succeeding classes are exhausted. The findings of this research reveals that the gcd of most of the pairs of the permuted classes are relatively prime. This results can be used further in constructing some cryptographic architectures that could be used in design of strong encryption schemes.

scholarly journals A Cryptographic System Based on a New Class of Binary Error-Correcting Codes

2019 ◽  
Vol 73 (1) ◽  
pp. 83-96
Author(s):  
Pál Dömösi ◽  
Carolin Hannusch ◽  
Géza Horváth
Keyword(s):  
Coding Theory ◽  
Error Correcting Code ◽  
Discrete Math ◽  
Error Correcting Codes ◽  
Key Generation ◽  
New Class ◽  
Algebraic Coding Theory ◽  
Encryption And Decryption ◽  
Construction Of Self ◽  
Cryptographic System

Abstract In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22k, 22k−1, 2k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

New entanglement-assisted quantum MDS codes

2021 ◽  
pp. 2150016
Author(s):  
Binbin Pang ◽  
Shixin Zhu ◽  
Liqi Wang
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Maximum Distance Separable ◽  
Number Formula ◽  
Minimum Distances ◽  
Quantum Error ◽  
Quantum Mds Codes

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.

Some results concerning linear codes and (k, 3)-caps in three-dimensional Galois space

1978 ◽  
Vol 84 (2) ◽  
pp. 191-205 ◽  
Author(s):  
Raymond Hill
Keyword(s):  
Upper Bound ◽  
Coding Theory ◽  
Linear Codes ◽  
Three Dimensional ◽  
Packing Problem ◽  
Error Correcting Codes ◽  
Finite Geometries ◽  
Weight Distributions ◽  
Galois Space

AbstractThe packing problem for (k, 3)-caps is that of finding (m, 3)r, q, the largest size of (k, 3)-cap in the Galois space Sr, q. The problem is tackled by exploiting the interplay of finite geometries with error-correcting codes. An improved general upper bound on (m, 3)3 q and the actual value of (m, 3)3, 4 are obtained. In terms of coding theory, the methods make a useful contribution to the difficult task of establishing the existence or non-existence of linear codes with certain weight distributions.

scholarly journals A survey on single server private information retrieval in a coding theory perspective

2021 ◽  
Author(s):  
Gianira N. Alfarano ◽  
Karan Khathuria ◽  
Violetta Weger
Keyword(s):  
Information Retrieval ◽  
Private Information ◽  
Coding Theory ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Point Of View ◽  
Single Server ◽  
Private Information Retrieval ◽  
Generic Framework ◽  
New Perspective

AbstractIn this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. This generic framework provides an appropriate setup to analyze the security of such PIR schemes. In fact, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a unified point of view.

scholarly journals Minimizing the Cost of Guessing Games

2018 ◽  
Vol 15 (2) ◽  
Author(s):  
David Clark ◽  
Lindsay Czap
Keyword(s):  
Cost Function ◽  
Coding Theory ◽  
Minimum Cost ◽  
Error Correcting Codes ◽  
Triple Systems ◽  
Pairwise Balanced Designs ◽  
Constant Cost ◽  
Balanced Designs ◽  
The Cost ◽  
Guessing Games

A two-player “guessing game” is a game in which the first participant, the “Responder,” picks a number from a certain range. Then, the second participant, the “Questioner,” asks only yes-or-no questions in order to guess the number. In this paper, we study guessing games with lies and costs. In particular, the Responder is allowed to lie in one answer, and the Questioner is charged a cost based on the content of each question. Guessing games with lies are closely linked to error correcting codes, which are mathematical objects that allow us to detect an error in received information and correct these errors. We will give basic definitions in coding theory and show how error correcting codes allow us to still guess the correct number even if one lie is involved. We will additionally seek to minimize the total cost of our games. We will provide explicit constructions, for any cost function, for games with the minimum possible cost and an unlimited number of questions. We also find minimum cost games for games with a restricted number of questions and a constant cost function. KEYWORDS: Ulam’s Game; Guessing Games With Lies; Error Correcting Codes; Pairwise Balanced Designs; Steiner Triple Systems

scholarly journals On the Failing Cases of the Johnson Bound for Error-Correcting Codes

10.37236/779 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Wolfgang Haas
Keyword(s):  
Coding Theory ◽  
Sphere Packing ◽  
Steiner System ◽  
Error Correcting Codes ◽  
Positive Density ◽  
System Of Linear Inequalities ◽  
Binary Case ◽  
Johnson Bound ◽  
Hamming Space ◽  
The Difference

A central problem in coding theory is to determine $A_q(n,2e+1)$, the maximal cardinality of a $q$-ary code of length $n$ correcting up to $e$ errors. When $e$ is fixed and $n$ is large, the best upper bound for $A(n,2e+1)$ (the binary case) is the well-known Johnson bound from 1962. This however simply reduces to the sphere-packing bound if a Steiner system $S(e+1,2e+1,n)$ exists. Despite the fact that no such system is known whenever $e\geq 5$, they possibly exist for a set of values for $n$ with positive density. Therefore in these cases no non-trivial numerical upper bounds for $A(n,2e+1)$ are known. In this paper the author demonstrates a technique for upper-bounding $A_q(n,2e+1)$, which closes this gap in coding theory. The author extends his earlier work on the system of linear inequalities satisfied by the number of elements of certain codes lying in $k$-dimensional subspaces of the Hamming Space. The method suffices to give the first proof, that the difference between the sphere-packing bound and $A_q(n,2e+1)$ approaches infinity with increasing $n$ whenever $q$ and $e\geq 2$ are fixed. A similar result holds for $K_q(n,R)$, the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Moreover the author presents a new bound for $A(n,3)$ giving for instance $A(19,3)\leq 26168$.

Sign in / Sign up

Export Citation Format

Share Document