The Walsh transform of a class of monomial functions and cyclic codes

Chengju Li; Qin Yue

doi:10.1007/s12095-014-0109-2

The Walsh transform of a class of monomial functions and cyclic codes

Chengju Li^*
, Qin Yue

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

Let 𝔽_p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying p^f ≡ −1 (mod N). Then we let q = p^2f and r = q^m. In this paper, we study the Walsh transform of the monomial function.

(Formula presented.).

for (Formula presented.). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most (Formula presented.) distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.

Original language	English
Pages (from-to)	217-228
Number of pages	12
Journal	Cryptography and Communications
Volume	7
Issue number	2
DOIs	https://doi.org/10.1007/s12095-014-0109-2
State	Published - Jun 2014
Externally published	Yes

Keywords

Cyclic code
Gauss periods
Gauss sums
Value distribution
Walsh transform

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10.1007/s12095-014-0109-2

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title = "The Walsh transform of a class of monomial functions and cyclic codes",

abstract = "Let 𝔽p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying pf ≡ −1 (mod N). Then we let q = p2f and r = qm. In this paper, we study the Walsh transform of the monomial function.(Formula presented.).for (Formula presented.). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most (Formula presented.) distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.",

keywords = "Cyclic code, Gauss periods, Gauss sums, Value distribution, Walsh transform",

author = "Chengju Li and Qin Yue",

note = "Publisher Copyright: {\textcopyright} 2014, Springer Science+Business Media New York.",

year = "2014",

month = jun,

doi = "10.1007/s12095-014-0109-2",

language = "英语",

volume = "7",

pages = "217--228",

journal = "Cryptography and Communications",

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TY - JOUR

T1 - The Walsh transform of a class of monomial functions and cyclic codes

AU - Li, Chengju

AU - Yue, Qin

PY - 2014/6

Y1 - 2014/6

N2 - Let 𝔽p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying pf ≡ −1 (mod N). Then we let q = p2f and r = qm. In this paper, we study the Walsh transform of the monomial function.(Formula presented.).for (Formula presented.). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most (Formula presented.) distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.

AB - Let 𝔽p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying pf ≡ −1 (mod N). Then we let q = p2f and r = qm. In this paper, we study the Walsh transform of the monomial function.(Formula presented.).for (Formula presented.). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most (Formula presented.) distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.

KW - Cyclic code

KW - Gauss periods

KW - Gauss sums

KW - Value distribution

KW - Walsh transform

UR - https://www.scopus.com/pages/publications/84921932265

U2 - 10.1007/s12095-014-0109-2

DO - 10.1007/s12095-014-0109-2

M3 - 文章

AN - SCOPUS:84921932265

SN - 1936-2447

VL - 7

SP - 217

EP - 228

JO - Cryptography and Communications

JF - Cryptography and Communications

IS - 2

ER -

The Walsh transform of a class of monomial functions and cyclic codes

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Keywords

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