scholarly journals Codes and Projective Multisets

10.37236/1375 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stefan Dodunekov ◽  
Juriaan Simonis
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Full Length ◽  
Self Duality ◽  
Projective Codes ◽  
Matrix Free

The paper gives a matrix-free presentation of the correspondence between full-length linear codes and projective multisets. It generalizes the Brouwer-Van Eupen construction that transforms projective codes into two-weight codes. Short proofs of known theorems are obtained. A new notion of self-duality in coding theory is explored.

scholarly journals Codes on Key Errors

2014 ◽  
Vol 14 (2) ◽  
pp. 31-37 ◽  
Author(s):  
P. K. Das
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Simultaneous Detection ◽  
Communication Channels ◽  
Upper Bounds ◽  
Parity Check ◽  
Lower And Upper Bounds ◽  
Different Types

Abstract Coding theory has started with the intention of detection and correction of errors which have occurred during communication. Different types of errors are produced by different types of communication channels and accordingly codes are developed to deal with them. In 2013 Sharma and Gaur introduced a new kind of an error which will be termed “key error”. This paper obtains the lower and upper bounds on the number of parity-check digits required for linear codes capable for detecting such errors. Illustration of such a code is provided. Codes capable of simultaneous detection and correction of such errors have also been considered.

Some results on quadratic residue codes over the ring Fp + vFp + v2Fp + v3Fp

2017 ◽  
Vol 09 (03) ◽  
pp. 1750035
Author(s):  
Jian Gao ◽  
Fanghui Ma
Keyword(s):  
Linear Codes ◽  
Quadratic Residue ◽  
The Self ◽  
Dual Code ◽  
Gray Map ◽  
Chain Ring ◽  
Qr Codes ◽  
Quadratic Residue Codes ◽  
Self Duality ◽  
Gray Maps

Quadratic residue (QR) codes and their extensions over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text]. A class of Gray maps preserving the self-duality of linear codes from [Formula: see text] to [Formula: see text] is given. Under a special Gray map, a self-dual code [Formula: see text] over [Formula: see text], a formally self-dual code [Formula: see text] over [Formula: see text] and a formally self-dual code [Formula: see text] over [Formula: see text] are obtained from extended QR codes.

REPEATED BURST ERROR CORRECTING LINEAR CODES

2008 ◽  
Vol 01 (03) ◽  
pp. 303-335 ◽  
Author(s):  
B. K. Dass ◽  
Rashmi Verma
Keyword(s):  
Data Transmission ◽  
Coding Theory ◽  
Linear Codes ◽  
Practical Importance ◽  
Practical Interest ◽  
Lower And Upper Bounds ◽  
Burst Error ◽  
The Subject ◽  
Error Detecting ◽  
Parallel Growth

Many kinds of errors in coding theory have been dealt with for which codes have been constructed to combat such errors. Though there is a long history concerning the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest, one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. The nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of data transmission. In very busy communication channels, errors repeat themselves more frequently. In view of this, it is desirable to consider repeated burst errors. The paper presents lower and upper bounds on the number of parity-check digits required for a linear code correcting errors in the form of repeated bursts. An upper bound for a code that detects m-repeated bursts has also been derived. Illustrations of several codes that correct 2-repeated bursts of different lengths have also been given.

scholarly journals Infinite Families of 2-Designs from a Class of Linear Codes Related to Dembowski-Ostrom Functions

2021 ◽  
pp. 1-15
Author(s):  
Rong Wang ◽  
Xiaoni Du ◽  
Cuiling Fan ◽  
Zhihua Niu
Keyword(s):  
Finite Field ◽  
Weight Distribution ◽  
Coding Theory ◽  
Exponential Sums ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Research Interest ◽  
Combinatorial Formula ◽  
Text Design ◽  
Fixed Weight

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial [Formula: see text]-designs have attracted lots of research interest for decades. The interplay between coding theory and [Formula: see text]-designs started many years ago. It is generally known that [Formula: see text]-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a [Formula: see text]-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of [Formula: see text]-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of [Formula: see text]-designs are calculated explicitly.

scholarly journals Counting Z2 Z4 Z8-additive Codes

2019 ◽  
Vol 12 (2) ◽  
pp. 668-679
Author(s):  
Basri Çalışkan ◽  
Kemal Balıkçı
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Algebraic Coding Theory ◽  
Additive Codes ◽  
Generator Matrices

In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.

scholarly journals Partitions on the Projective Line and Plane of Order 17k,k=1,2,3

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm A.M. Al-Seraji ◽  
Hossam H. Jawad
Keyword(s):  
Projective Plane ◽  
Coding Theory ◽  
Projective Line ◽  
Projective Codes

In this research, the main purposes are making partitions of the projective line, PG (1, q3), q=17, and embedding the projective line, PG (1, q ) into PG (1, q3), as well as finding some partitions of PG (2, q2) as subplanes, orbits, triangles and arcs, and studying the properties of these subsets. Furthermore, the arcs with different degrees and sizes are found besides the embedding of the projective plane, PG (2,q) into PG (2, q2). In the coding theory, there are 14 projective codes that are introduced.

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.

scholarly journals Designing Efficient Dyadic Operations for Cryptographic Applications

2020 ◽  
Vol 14 (1) ◽  
pp. 95-109
Author(s):  
Gustavo Banegas ◽  
Paulo S. L. M. Barreto ◽  
Edoardo Persichetti ◽  
Paolo Santini
Keyword(s):  
Automorphism Group ◽  
Quantum Cryptography ◽  
Coding Theory ◽  
Linear Codes ◽  
Key Exchange ◽  
Independent Interest ◽  
Symmetric Matrices ◽  
Cryptographic Primitives ◽  
Standardization Process ◽  
Post Quantum Cryptography

AbstractCryptographic primitives from coding theory are some of the most promising candidates for NIST’s Post-Quantum Cryptography Standardization process. In this paper, we introduce a variety of techniques to improve operations on dyadic matrices, a particular type of symmetric matrices that appear in the automorphism group of certain linear codes. Besides the independent interest, these techniques find an immediate application in practice. In fact, one of the candidates for the Key Exchange functionality, called DAGS, makes use of quasi-dyadic matrices to provide compact keys for the scheme.

scholarly journals Codes correcting and simultaneously detecting solid burst errors

2013 ◽  
Vol 2 (1) ◽  
pp. 143-150
Author(s):  
P.K. Das
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Communication Channels ◽  
Upper Bounds ◽  
Parity Check ◽  
Lower And Upper Bounds ◽  
Parity Check Matrix ◽  
Burst Error ◽  
Check Matrix ◽  
Error Detecting

Detecting and correcting errors is one of the main tasks in coding theory. The bounds are important in terms of error-detecting and -correcting capabilities of the codes. Solid Burst error is common in several communication channels. This paper obtains lower and upper bounds on the number of parity-check digits required for linear codes capable of correcting any solid burst error of length b or less and simultaneously detecting any solid burst error of length s(>b) or less. Illustration of such a code is also provided.Keywords: Parity check matrix, Syndromes, Solid burst errors, Standard arrayDOI: 10.18495/comengapp.21.143150  

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.

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