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Cryptography and Communications

Erschienen in:

01.06.2015

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

verfasst von: Chengju Li, Qin Yue

Erschienen in: Cryptography and Communications | Ausgabe 2/2015

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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

\(f(x)=\text {Tr}_{r/p}(ax^{\frac {r-1} N})\)

for \(a \in \Bbb F_{r}^{*}\). We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most \(\min \{p, N\}+1\) 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.

Vorheriger Artikel A construction of t-fold perfect splitting authentication codes with equal deception probabilities

Nächster Artikel Repeated-root constacyclic codes of length ℓ t p s and their dual codes

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Titel: The Walsh transform of a class of monomial functions and cyclic codes
verfasst von: Chengju Li
Qin Yue
Publikationsdatum: 01.06.2015
Verlag: Springer US
Erschienen in: Cryptography and Communications / Ausgabe 2/2015
Print ISSN: 1936-2447
Elektronische ISSN: 1936-2455
DOI: https://doi.org/10.1007/s12095-014-0109-2

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