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

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.

Download Full-text

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055031 ◽

1978 ◽

Vol 84 (2) ◽

pp. 191-205 ◽

Cited By ~ 5

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.

Download Full-text

Weight distributions for projective binary linear codes from Weil sums

AIMS Mathematics ◽

10.3934/math.2021499 ◽

2021 ◽

Vol 6 (8) ◽

pp. 8600-8610

Author(s):

Shudi Yang ◽

◽

Zheng-An Yao ◽

Keyword(s):

Linear Codes ◽

Binary Linear Codes ◽

Weight Distributions ◽

Weil Sums

Download Full-text

More Constructions of 3-Weight Linear Codes

Journal of Mathematics ◽

10.1155/2021/6662843 ◽

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.

Download Full-text

On modules with cyclic socle

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501620 ◽

2016 ◽

Vol 15 (09) ◽

pp. 1650162 ◽

Cited By ~ 1

Author(s):

Ali Assem

Keyword(s):

Coding Theory ◽

Linear Codes ◽

Extension Theorem ◽

Extension Property ◽

Codes Over Rings ◽

Extension Problem ◽

Hamming Weight ◽

Frobenius Rings ◽

Code Equivalence ◽

Open Question

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

Download Full-text