scholarly journals More Constructions of 3-Weight Linear Codes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lingyong Ma ◽  
Guanjun Li ◽  
Fengyan Liu
Keyword(s):  
Coding Theory ◽  
Exponential Sums ◽  
Linear Codes ◽  
Research Topic ◽  
Plateaued Functions ◽  
Weight Distributions

Linear codes with few weights have become an interesting research topic and important applications of cryptography and coding theory. In this paper, we apply some ternary near-bent and 2-plateaued functions or r -ary functions to construct more 3-weight linear codes, where r is a prime. Moreover, we determine the weight distributions of the resulted linear codes by means of some exponential sums.

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.

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 On the most Weight $w$ Vectors in a Dimension $k$ Binary Code

10.37236/414 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Joshua Brown Kramer
Keyword(s):  
Coding Theory ◽  
Binary Code ◽  
Linear Codes ◽  
Complete Solution ◽  
Dimensional Subspace ◽  
Hamming Weight ◽  
Binary Linear Codes ◽  
Linear Map ◽  
Weight Distributions

Ahlswede, Aydinian, and Khachatrian posed the following problem: what is the maximum number of Hamming weight $w$ vectors in a $k$-dimensional subspace of $\mathbb{F}_2^n$? The answer to this question could be relevant to coding theory, since it sheds light on the weight distributions of binary linear codes. We give some partial results. We also provide a conjecture for the complete solution when $w$ is odd as well as for the case $k \geq 2w$ and $w$ even. One tool used to study this problem is a linear map that decreases the weight of nonzero vectors by a constant. We characterize such maps.

Several classes of linear codes and their weight distributions

2018 ◽  
Vol 30 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Xiaoqiang Wang ◽  
Dabin Zheng ◽  
Hongwei Liu
Keyword(s):  
Linear Codes ◽  
Weight Distributions

Linear codes with few weights from weakly regular plateaued functions

2021 ◽  
Vol 344 (12) ◽  
pp. 112597
Author(s):  
Yingjie Cheng ◽  
Xiwang Cao
Keyword(s):  
Linear Codes ◽  
Plateaued Functions

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.

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.

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