scholarly journals A class of the hamming weight hierarchy of linear codes with dimension 5

2014 ◽  
Vol 19 (5) ◽  
pp. 442-451
Author(s):  
Guoxiang Hu ◽  
Huanguo Zhang ◽  
Lijun Wang ◽  
Zhe Dong
Keyword(s):  
Linear Codes ◽  
Hamming Weight ◽  
Weight Hierarchy

scholarly journals THE WEIGHT HIERARCHY OF HADAMARD CODES

2019 ◽  
pp. 797
Author(s):  
Farzaneh Farhang Baftani ◽  
Hamid Reza Maimani
Keyword(s):  
Binary Code ◽  
Hadamard Matrix ◽  
Hamming Weight ◽  
Generalized Hamming Weight ◽  
Hadamard Codes ◽  
Hadamard Code ◽  
Weight Hierarchy

The support of an $(n, M, d)$ binary code  $C$ over the set $\mathbf{A}=\{0,1\}$ is the set of all coordinate positions $i$, such  that  at  least two codewords  have distinct entry  in  coordinate $i$.  The  $r$th  generalized  Hamming  weight  $d_r(C)$,  $1\leq r\leq 1+log_2n+1$,  of  $C$  is  defined  as  the minimum  of  the  cardinalities  of  the  supports  of  all subset  of  $C$ of cardinality $2^{r-1}+1$.  The  sequence $(d_1(C), d_2(C), \ldots, d_k(C))$ is called the Hamming weight hierarchy (HWH) of $C$. In this paper we obtain HWH for $(2^k-1, 2^k, 2^{k-1}$ binary Hadamard code corresponding to Sylvester Hadamard matrix $H_{2^k}$ and we show that    $$d_r=2^{k-r} (2^r -1).$$ Also we compute the HWH of all $(4n-1, 4n, 2n)$ Hadamard code for $2\leq n\leq 8$.

On cyclic codes in incidence rings

2006 ◽  
Vol 43 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Ricardo Alfaro ◽  
Andrei V. Kelarev
Keyword(s):  
Directed Graph ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Matrix Ring ◽  
Hamming Weight ◽  
Quotient Rings

Cyclic codes are defined as ideals in polynomial quotient rings. We are using a matrix ring construction in a similar way to define classes of codes. It is shown that all cyclic and all linear codes can be embedded as ideals in this construction. A formula for the largest Hamming weight of one-sided ideals in incidence rings is given. It is shown that every incidence ring defined by a directed graph always possesses a principal one-sided ideal that achieves the optimum Hamming weight.

scholarly journals Type IV codes over a non-unital ring

2021 ◽  
pp. 2250142
Author(s):  
Adel Alahmadi ◽  
Alaa Altassan ◽  
Widyan Basaffar ◽  
Hatoon Shoaib ◽  
Alexis Bonnecaze ◽  
...  
Keyword(s):  
Local Ring ◽  
Algebraic Structure ◽  
Invariant Theory ◽  
Linear Codes ◽  
Hamming Weight ◽  
Dual Codes ◽  
Weight Enumerators ◽  
Type Iv ◽  
Commutative Local Ring ◽  
Torsion Codes

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

scholarly journals Certain Types of Linear Codes over the Finite Field of Order Twenty-Five

2021 ◽  
pp. 4019-4031
Author(s):  
Emad Bakr Al-Zangana ◽  
Elaf Abdul Satar Shehab
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Weight Distribution ◽  
Incidence Matrix ◽  
Linear Codes ◽  
Projective Line ◽  
Three Dimensions ◽  
Hamming Weight ◽  
Maximum Distance Separable ◽  
Maximum Distance Separable Codes

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of  were studied over different finite fields.  

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.

On modules with cyclic socle

2016 ◽  
Vol 15 (09) ◽  
pp. 1650162 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document