Journal ArticleDOI
The covering radius of the (2^{15}, 16) Reed-Muller code is at least 16276
N. Patterson,D. Wiedemann +1 more
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TLDR
Two interesting cosets of the first-order Reed-Muller code of block length 2^{15} are described, which provide counterexamples to a conjecture on the covering radius.Abstract:
Two interesting cosets of the first-order Reed-Muller code of block length 2^{15} are described. They provide counterexamples to a conjecture on the covering radius.read more
Citations
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Book ChapterDOI
Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes
TL;DR: Encryption-decryption is the most ancient cryptographic activity, but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power.
Journal ArticleDOI
Covering radius---Survey and recent results
TL;DR: A number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, and some extensive relations with other aspects of coding theory through the Reed-Muller codes are presented.
Journal Article
Construction of bent functions and balanced Boolean functions with high nonlinearity
TL;DR: In this article, a general explicit construction of bent functions is described, which unifies well known constructions due to Maiorana-McFarland and Dillon as two opposite extremal cases.
Journal ArticleDOI
On the covering radius of codes
Ron Graham,Neil J. A. Sloane +1 more
TL;DR: This work gives a number of new results concerning t[n, k] , the minimal covering radius of any binary code of length n and dimension k, which are determined exactly and reasonably tight bounds are obtained for any k when n is large.
Journal ArticleDOI
A new class of zero-correlation zone sequences
TL;DR: Two new methods for constructing sets of zero-correlation zone (ZCZ) sequences are proposed, based on perfect sequences and unitary matrices, which are suitable for approximately synchronized code-division multiple-access (AS-CDMA) systems.
References
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Journal ArticleDOI
On “bent” functions
TL;DR: The polynomial degree of a bent function P ( x ) is studied, as are the properties of the Fourier transform of (−1) P(x) , and a connection with Hadamard matrices.
Journal ArticleDOI
Weight distributions of the cosets of the (32,6) Reed-Muller code
Elwyn R. Berlekamp,L. R. Welch +1 more
TL;DR: The weight distribution of all 2^26 cosets of the (32,6) first-order Reed-Muller code is presented, and this equivalent problem: how well are the 2^32 Boolean functions of five variables approximated by the2^5 linear functions and their complements?
Journal ArticleDOI
On the covering radius of binary codes (Corresp.)
TL;DR: It is shown that the covering radius r_{m} of the first-order Reed-Muller code of lenglh 2^{m} satisfies 2-2 m-l-2 lceil m/2 r-rceil -1 r-m, which is equivalent to 2.m-1-2.m/2-1.
Journal ArticleDOI
The covering radius of the (128,8) Reed-Muller code is 56 (Corresp.)
TL;DR: In this paper, it was shown that the Reed-Muller code holds for all m = 0, 1, 2, and for m = 3, where m is the covering radius of the code.