Combinatorial t -designs from special functions
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Combinatorial t -designs from special functions Cunsheng Ding1 · Chunming Tang2 Received: 4 September 2019 / Accepted: 4 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions, and planar functions. These combinatorial designs are not constructed directly from such functions, but come from linear codes which are constructed with such functions. As a byproduct, this paper also surveys linear codes from certain special functions. Keywords Cyclic code · Design · Linear code · Special function Mathematics Subject Classification 2010 05B05
1 Introduction Let P be a set of v ≥ 1 elements, and let B be a set of k-subsets of P , where k is a positive integer with 1 ≤ k ≤ v. Let t be a positive integer with t ≤ k. The pair D = (P , B ) is called a t-(v, k, λ) design, or simply t-design, if every t-subset of P is contained in exactly λ elements of B . The elements of P are called points, and those of B are referred to as This article belongs to the Topical Collection: Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds). Chunming Tang
[email protected] Cunsheng Ding [email protected] 1
Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
2
The School of Mathematics and Information, China West Normal University, Nanchong 637002, China
Cryptography and Communications
blocks. We usually use b to denote the number of blocks in B . A t-design is called simple if B does not contain repeated blocks. In this survey, we consider only simple t-designs. A t-design is called symmetric if v = b. It is clear that t-designs with k = t or k = v always exist. Such t-designs are trivial. In this survey, we consider only t-designs with v > k > t. A t-(v, k, λ) design is referred to as a Steiner system if t ≥ 2 and λ = 1, and is denoted by S(t, k, v). We assume that the reader is familiar with the basics of linear codes and cyclic codes, and proceed to introduce the classical construction of t-designs from codes directly. Let C be a [v, κ, d] linear code over GF(q). Let Ai := Ai (C), which denotes the number of codewords with Hamming weight i in C, where 0 ≤ i ≤ v. The sequence (A0 , A1 , · · · , Av ) is called the weight distribution of C, and vi=0 Ai zi is referred to as the weight enumerator of C. For e
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