Algorithmic complexity in coding theory and the minimum distance problem

1997 ◽  
Author(s):  
Alexander Vardy
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Algorithmic Complexity ◽  
Distance Problem

A Maxmin Distance Problem

1972 ◽  
Vol 94 (2) ◽  
pp. 155-158 ◽  
Author(s):  
R. Aggarwal ◽  
G. Leitmann
Keyword(s):  
Minimum Distance ◽  
Terminal State ◽  
Initial State ◽  
Closed Set ◽  
Distance Problem

The problem of maximizing the minimum distance of a dynamical system’s state from a given closed set, while transferring the system from a given initial state to a given terminal state, is considered. Two different methods of solution of this problem are given.

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 Applications of the Projective Plane in Coding Theory

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Najm Abdulzahra Makhrib Al-Seraji ◽  
Zainab Sadiq Jafar
Keyword(s):  
Projective Plane ◽  
Coding Theory ◽  
Minimum Distance ◽  
Incidence Matrix ◽  
Galois Field ◽  
Programming System ◽  
Combinatorial Structures

The goal of this paper was to study the applications of the projective plane PG (2, q) over a Galois field of order q in the projective linear (n, k, d, q) -code such that the parameters length of code n, the dimension of code k, and the minimum distance d with the error-correcting e according to an incidence matrix have been calculated. Also, this research provides examples and theorems of links between the combinatorial structures and coding theory. The calculations depend on the GAP (groups, algorithms, and programming) system. The method of the research depends on the classification of the points and lines in PG (2, q).

scholarly journals Some Remarks on the Plotkin Bound

10.37236/1746 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Jörn Quistorff
Keyword(s):  
Upper Bound ◽  
Coding Theory ◽  
Minimum Distance ◽  
Binary Codes ◽  
Plotkin Bound ◽  
Hamming Metric ◽  
Lee Metric

In coding theory, Plotkin's upper bound on the maximal cadinality of a code with minimum distance at least $d$ is well known. He presented it for binary codes where Hamming and Lee metric coincide. After a brief discussion of the generalization to $q$-ary codes preserved with the Hamming metric, the application of the Plotkin bound to $q$-ary codes preserved with the Lee metric due to Wyner and Graham is improved.

scholarly journals Some Applications of Coding Theory in the Projective Plane of Order Four

2019 ◽  
Vol 30 (1) ◽  
pp. 152
Author(s):  
Najm A. M. AL-Seraji ◽  
Hamza L. M. Ajaj
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Minimum Distance ◽  
Incidence Matrix ◽  
Maximum Value ◽  
Order Four ◽  
The Relationship

The main aim of this research is to introduce the relationship between the topic of coding theory and the projective plane of order four. The maximum value of size code M over the finite field of order four and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. Some examples and theorems have been given.

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.

Using Genetic Algorithms With Niche Formation to Solve the Minimum Distance Problem Among Concave Objects

2002 ◽  
Author(s):  
J. A. Carretero ◽  
M. A. Nahon ◽  
O. Ma
Keyword(s):  
Minimum Distance ◽  
Optimization Technique ◽  
Combinatorial Problem ◽  
The Body ◽  
Optimization Approach ◽  
Multiple Minima ◽  
Niche Formation ◽  
Distance Problem ◽  
Set Of Points ◽  
Global Optimization Technique

In this paper an optimization approach is used to solve the problem of finding the minimum distance between concave objects, without the need for partitioning the objects into convex sub-objects. Since the optimization problem is not unimodal (i.e., has more than one local minimum point), a global optimization technique, namely a Genetic Algorithm, is used to solve the concave problem. In order to reduce the computational expense of evaluating the constraints at runtime, the objects’ geometry is replaced by a set of points on the surface of the body. This reduces the problem to a combinatorial problem where the combination of points (one on each body) that minimizes the distance will be the solution. Additionally, niche formation is used to allow the minimum distance algorithm to track multiple minima rather than exclusively looking for the global minimum.

Sign in / Sign up

Export Citation Format

Share Document