Q-ary Functions Research Articles

Minimal linear codes constructed from functions

In this paper, we consider minimal linear codes by a general construction of linear codes from q-ary functions. First, we give necessary and sufficient conditions for codewords which are constructed by functions to be minimal. Second, as applications, we present three constructions of minimal linear codes. Constructions on minimal linear codes in this paper generalize some recent results in Ding et al. (IEEE Trans. Inf. Theory 64(10), 6536–6545, 2018); Heng et al. (Finite Fields Appl. 54, 176–196, 2018); Bartoli and Bonini (IEEE Trans. Inf. Theory 65(7), 4152–4155, 2019); Mesnager et al. (IEEE Trans. Inf. Theory 66(9), 5404–5413, 2020); Bonini and Borello (J. Algebraic Comb. 53, 327–341, 2021). In our three constructions, the conditions of functions are much looser than theirs.

On [formula omitted]-ary plateaued functions over [formula omitted] and their explicit characterizations

Plateaued and bent functions play a significant role in cryptography, sequence theory, coding theory and combinatorics. In 1997, Coulter and Matthews redefined bent functions over any finite field Fq where q is a prime power, and established their properties. The objective of this work is to redefine the notion of plateaued functions over Fq, and to present several explicit characterizations of those functions. We first give, over Fq, the notion of q-ary plateaued functions, which relies on the concept of the Walsh–Hadamard transform in terms of canonical additive character of Fq. We then give a concrete example of q-ary plateaued function, that is not vectorial p-ary plateaued function. This suggests that the study of plateaued-ness is also significant for q-ary functions over Fq. We finally characterize q-ary plateaued functions in terms of derivatives, Walsh power moments and autocorrelation functions.

Constructions of q-ary functions with good global avalanche characteristics

In this paper, we construct two classes of q-ary balanced functions which have good global avalanche characteristics (GAC) measured in terms of sum-of-squares-modulus indicator (SSMI), modulus indicator(MI), and propagation criterion (PC). We show that the SSMI, MI, and PC of q-ary functions are invariant under affine transformations. Also, we give a construction of q-ary s-plateaued functions and obtain their SSMI. We provide a relationship between the autocorrelation spectrum of a cubic Boolean function and the dimension of the kernel of the bilinear form associated with the derivative of the function. Using this result, we identify several classes of cubic semi-bent Boolean functions which have good bounds on their SSMI and MI, and hence show good behaviour with respect to the GAC.

Some results on q-ary bent functions

Kumar et al. [Generalized bent functions and their properties, J. Comb. Theory Ser. A 40 (1985), pp. 90–107] have extended the notion of classical bent Boolean functions in the generalized setup on . They have provided an analogue of classical Maiorana-McFarland type bent functions. In this paper, we study the cross-correlation of a subclass of such generalized Maiorana-McFarland type bent functions. We provide a characterization of quaternary (q=4) bent functions on n+1 variables in terms of their subfunctions on n-variables. Analogues of sum-of-squares’ indicator and absolute indicator of cross-correlation of Boolean functions are defined in the generalized setup. Further, q-ary functions are studied in terms of these indicators and some upper bounds of these indicators are obtained. Finally, we provide some constructions of balanced quaternary functions with high nonlinearity under Lee metric.