New bounds on the covering radius of the second order Reed-Muller code of length 128
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Authors
Wang, Qichun
Stănică, Pantelimon
Subjects
Boolean functions
covering radius
Reed-Muller codes
second-order nonlinearity
covering radius
Reed-Muller codes
second-order nonlinearity
Advisors
Date of Issue
2019
Date
2019
Publisher
Language
en_US
Abstract
In 1981, Schatz proved that the covering radius of the binary Reed- Muller code RM (2, 6) is 18. It was previously shown that the covering radius of RM(2,7) is between 40 and 44. In this paper, we prove that the covering radius of RM(2,7) is at most 42. As a corollary, we also find new upper bounds for RM (2, n), n = 8, 9, 10. Moreover, we give a sufficient and necessary condition for the covering radius of RM(2,7) to be equal to 42. Using this condition, we prove that the covering radius of RM(2,7) in RM(4,7) is exactly 40, and as a by-product, we conclude that the covering radius of RM(2,7) in the set of 2-resilient Boolean functions is at most 40, which improves the bound given by Borissov et al. (IEEE Trans. Inf. Theory 51:1182-1189, 2005).
Type
Article
Description
Series/Report No
Department
Applied Mathematics
Organization
Naval Postgraduate School (U.S.)
Identifiers
NPS Report Number
Sponsors
The first author would like to thank the financial support from the National Natural Science Foundation of China (Grants 61572189 and 61202463).
Funder
The first author would like to thank the financial support from the National Natural Science Foundation of China (Grants 61572189 and 61202463).
Format
12 p.
Citation
Q. Wang, P. Stanica, A new upper bound for the covering radius of the second order Reed-Muller code of length 128, Cryptography and Communications, 11:2 (2019), 269-277.
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.