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THE WEIGHT HIERARCHY OF HADAMARD CODES

Farzaneh Farhang Baftani, Hamid Reza Maimani
2019 Facta Universitatis Series Mathematics and Informatics  
The sequence $(d_1(C), d_2(C), \ldots, d_k(C))$ is called the Hamming weight hierarchy (HWH) of $C$.  ...  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 $i$.  ...  Acknowledgments The authors would like to thank the referee for his/her helpful remarks which have contributed to improve the presentation of the article.  ... 
doi:10.22190/fumi1904797f fatcat:lgodlb2b3fhsrfjldg4dgdsdei

The weight hierarchy of product codes

H.G. Schaathun
2000 IEEE Transactions on Information Theory  
The weight of a code is the number of coordinate positions where no codeword is zero. The th minimum weight is the least weight of an -dimensional subcode.  ...  Index Terms-Product code, projective system (projective multiset), Segre embedding, weight hierarchy.  ...  hierarchies of the component codes.  ... 
doi:10.1109/18.887876 fatcat:7bdifbhudvanlabixjndmsheum

The weight hierarchy of the Kasami codes

Tor Helleseth, P Vijay Kumar
1995 Discrete Mathematics  
The weight hierarchy of the Kasami codes is completely determined. It is also shown that the chain condition holds for these codes.  ...  The Kasami codes is a family of [2 2m --1, 3m, 2 TM 1 _ 2"-1 ] codes which are well known for their applications to construct sequences with optimal correlation magnitudes.  ...  As pointed out to us by one of the referees, there is a p-ary generalization of the Kasami codes due to Delsarte and Goethals [2] .  ... 
doi:10.1016/0012-365x(94)00030-m fatcat:mintovcjefa2jnscbbrosorelq

The weight hierarchies of outer product codes

Jeng Yune Park
2000 Discrete Mathematics  
The weight hierarchy of the outer product of two self-complementary codes is expressed as a function of the weight hierarchies of the component codes.  ...  An alternative method to calculate the rth generalized Hamming weight (and so the weight hierarchy) of the ÿrst-order Reed-Muller code is given.  ...  In this paper, we study the relation between the weight hierarchy of an outer product code and the weight hierarchies of its component codes.  ... 
doi:10.1016/s0012-365x(00)00063-7 fatcat:dqj22z6agndsdmszmbwf7q2y7e

On the weight hierarchy of the semiprimitive codes

Tor Helleseth, P. Vijay Kumar
1996 Discrete Mathematics  
The complete weight hierarchy of the semiprimitive codes is determined when k/2j is odd.  ...  We apply the above results to find the generalized Hamming weight of some classes of dual codes of primitive BCH codes with designed distance N + 2 when k/2j is odd.  ...  Acknowledgements The authors would like to thank H. Stichtenoth for communicating his results in [4] to us which partly inspired the present paper.  ... 
doi:10.1016/0012-365x(94)00234-a fatcat:tlppzflyara5tdkcaikgrhabq4

The b-symbol weight hierarchy of the Kasami codes [article]

Hongwei Zhu, Minjia Shi
2021 arXiv   pre-print
It is motivated by the limitations of the reading process in high density data storage systems. One main task in b-symbol coding theory is to determine the b-symbol weight hierarchy of codes.  ...  In this paper, we study the b-symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes.  ...  The authors would like to thank Prof. T. Helleseth for helpful discussions.  ... 
arXiv:2112.04019v1 fatcat:r3ccvtogujft7aptozv6nddh5m

On the weight hierarchy of irreducible cyclic codes

Marcel van der Vlugt
1995 Journal of combinatorial theory. Series A  
We apply these methods to determine (parts of) the weight hierarchy of irreducible cyclic codes in the semiprimitive case.  ...  The weight hierarchy of c~ is strongly related to the weight hierarchy of the dual code cg± (see [ W]): {dr(C~):l<~r<~k}u{(n+l)-dr(~±):l<~r<~n-k}={1,2,...,n}. ( 2 . 1 ) 21 PROPOSITION.  ... 
doi:10.1016/0097-3165(95)90023-3 fatcat:lwsfxj32r5honl2sw7lw77rsx4

The Weight Hierarchy of Some Reducible Cyclic Codes [article]

Maosheng Xiong, Shuxing Li, Gennian Ge
2015 arXiv   pre-print
In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases.  ...  The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications.  ...  In this paper, we study the GHWs of a family of reducible codes introduced in [55] and obtain the weight hierarchy in several cases.  ... 
arXiv:1504.01274v1 fatcat:gpnwypr7dzeexn5adp6tyi26uu

On the Weight Hierarchy of Locally Repairable Codes [article]

Jie Hao, Shu-Tao Xia, Bin Chen, Fang-Wei Fu
2017 arXiv   pre-print
For an optimal (n,k,r) LRC with r ∤ k, we give lower bounds on the GHWs of the LRC and its dual code. Finally, two general bounds on linear codes in terms of the GHWs are presented.  ...  Then, it is shown that for an optimal (n,k,r) LRC with r | k, its weight hierarchy can be completely determined, and the i-th GHW of an optimal (n,k,r) LRC with r | k attains the proposed generalized Singleton-like  ...  The weight hierarchy of C is the set of integers {d i | 1 ≤ i ≤ k}. For an [n, k] linear code C, 1 ≤ d 1 < d 2 < · · · < d k = n.  ... 
arXiv:1702.05848v1 fatcat:3fv5lmncpzba7p5fg4lvhprtmu

Symmetric functions, codes of partitions and the KP hierarchy [article]

S.R. Carrell, I.P. Goulden
2009 arXiv   pre-print
As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plucker relations for Schur function coefficients in a tau function for the hierarchy  ...  This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition.  ...  in terms of codes of partitions.  ... 
arXiv:0902.4441v1 fatcat:gbcjioadrvam3ctorcd6heknvi

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes [article]

Thorsten Hehn, Olgica Milenkovic, Stefan Laendner, Johannes B. Huber
2007 arXiv   pre-print
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques.  ...  We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond  ...  But whenever such sets exist, they can be related to the stopping redundancy hierarchy and PD sets of the code.  ... 
arXiv:cs/0702050v2 fatcat:l5zt3aey5rcnlcgfbn23sqe6xi

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

Thorsten Hehn, Olgica Milenkovic, Stefan Laendner, Johannes B. Huber
2008 IEEE Transactions on Information Theory  
We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel  ...  Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.  ...  We extend the scope of the work in [10] by introducing the notion of the stopping redundancy hierarchy of a code.  ... 
doi:10.1109/tit.2008.2006456 fatcat:hv6cnyqt2fbotgscbx462ebfny

Symmetric functions, codes of partitions and the KP hierarchy

S. R. Carrell, I. P. Goulden
2009 Journal of Algebraic Combinatorics  
This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition.  ...  As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ -function for the hierarchy  ...  terms of codes of partitions.  ... 
doi:10.1007/s10801-009-0211-2 fatcat:wufux7vl3vbidnjmg65rir62re

The Weight Hierarchy of a Family of Cyclic Codes with Arbitrary Number of Nonzeroes [article]

Shuxing Li
2017 arXiv   pre-print
In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes.  ...  The weight hierarchy is determined by employing a number-theoretic approach.  ...  Acknowledgement The author wishes to express his gratitude to Professor Cunsheng Ding and Professor Maosheng Xiong, for their guidance and encouragement during his stay at the Hong Kong University of Science  ... 
arXiv:1702.01309v1 fatcat:eqjbbxh46febzil5l2mlzfzwey

On the Duality and File Size Hierarchy of Fractional Repetition Codes [article]

Bing Zhu, Kenneth W. Shum, Hui Li
2018 arXiv   pre-print
Moreover, we present the tensor product technique for combining FR codes, and elaborate on the file size hierarchy of resulting codes.  ...  We study the duality of FR codes, and investigate the relationship between the supported file size of an FR code and its dual code.  ...  THE HIERARCHY OF SUPPORTED FILE SIZE AND THE DUAL BOUND A. File Size Hierarchy of FR Codes Suppose that C = (P, B, I) is an (n, α, θ, ρ)-FR code.  ... 
arXiv:1808.01933v1 fatcat:lx2u55dbsrbydb33zkyszvvfl4
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