Generalized hamming weights of linear codes from cryptographic functions

Kangquan Li; Hao Chen; Longjiang Qu

doi:10.3934/amc.2022081

Article Contents

2022, Volume 16, Issue 4: 859-877. Doi: 10.3934/amc.2022081

This issue Previous Article Another expression of the MacWilliams identities and its applications Next Article A class of minimum storage cooperative regenerating codes with low access property

Generalized hamming weights of linear codes from cryptographic functions

1.
College of Science, National University of Defense Technology, Changsha 410073, China
2.
Hunan Engineering Research Center of Commercial Cryptography Theory, and Technology Innovation Changsha 410073, China
3.
College of Information Science and Technology/Cyber Security, Jinan University, Guangzhou 510632, China

^*Corresponding author: Longjiang Qu
^*Corresponding author: Longjiang Qu

Received: March 2022

Revised: September 2022

Early access: October 2022

Published: November 2022

The first author is supported by the National Natural Science Foundation of China (NSFC) under Grant 62202476 and the Research Fund of National University of Defense Technology under Grant ZK22-14. The second and third authors are supported by the National Natural Science Foundation of China (NSFC) under Grant 62032009

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Abstract

The generalized Hamming weights of linear codes have attracted scholars' attention since Wei used them to characterize the cryptography performance of a linear code over the wire-tap channel of type II in 1991. Generally speaking, it is hard to determine linear codes' generalized Hamming weights, especially the weight hierarchy. On the other hand, since Ding and Niederreiter presented a generic approach to construct linear codes with good properties by suitable defining sets in 2007, many linear codes with few weights have been obtained using cryptographic functions. However, there does not seem to be much research on the generalized Hamming weights of linear codes from cryptographic functions.

In this paper, we first provide a general formula to compute the generalized Hamming weights of linear codes from the defining sets, which generalizes several known results. Next, some weight hierarchies of linear codes from three famous classes of Bent functions (the Maiorana-McFarland class, the class $\mathcal{PS}_{ap}$ , the class $H$ ) and two classes of Semi-Bent functions (the Maiorana-McFarland class, the summation of $\mathcal{PS}_{ap}$ and $H$ ) are considered. We hope this paper can attract interested scholars to find more results about the generalized Hamming weights of linear codes from cryptographic functions.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 11T71.

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