Yue Zhou’s research while affiliated with National University of Defense Technology and other places

Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line ${{\,\mathrm{{PG}}\,}}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over ${{\,\mathrm{{PG}}\,}}(1,q^n)$ for any even $n\ge 6$ and odd q. In particular, the relevant family contains at least inequivalent members for given $q=p^r$ and $n=2t>8$, where $p=\textrm{char}({{\mathbb {F}}}_q)$. This is a great improvement of previous results: for given q and $n>8$, the number of inequivalent maximum scattered linear sets of ${{\,\mathrm{{PG}}\,}}(1,q^n)$ in all classes known so far, is smaller than $q^2\phi (n)/2$, where $\phi$ denotes Euler’s totient function. Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.

In this paper, we present two new families of APN functions. The first family is in bivariate form $\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$ over ${\mathbb F}_{2^{m}}^{2}$ . It is obtained by adding certain terms of the form $\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$ to a family of APN functions recently proposed by Gölo&gcaron;lu. The $\vphantom {_{\int _{\int }}}$ second family has the form $L(z)^{2^{m}+1}+vz^{2^{m}+1}$ over ${\mathbb F}_{{2^{3m}}}$ , which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the $\Gamma$ -rank of the constructed APN functions over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively.

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with 22m elements, where m is even. We obtain this result by proving that the APN functions introduced by Pott and the second author [22], which depend on three parameters k, s and α, are pairwise inequivalent for distinct choices of the parameters k and s. Moreover, we determine the automorphism group of these APN functions.

In this paper, we present two new infinite classes of APN functions over \gf_{{2^{2m}}} and \gf_{{2^{3m}}}, respectively. The first one is with bivariate form and obtained by adding special terms, $\sum(a_ix^{2^i}y^{2^i},b_ix^{2^i}y^{2^i})$, to a known class of APN functions by {G{\"{o}}lo{\v{g}}lu} over \gf_{{2^m}}^2. The second one is of the form $L(z)^{2^m+1}+vz^{2^m+1}$ over \gf_{{2^{3m}}}, which is a generalization of one family of APN functions by Bracken et al. [Cryptogr. Commun. 3 (1): 43-53, 2011]. The calculation of the CCZ-invariants $\Gamma$-ranks of our APN classes over \gf_{{2^8}} or \gf_{{2^9}} indicates that they are CCZ-inequivalent to all known infinite families of APN functions. Moreover, by using the code isomorphism, we see that our first APN family covers an APN function over \gf_{{2^8}} obtained through the switching method by Edel and Pott in [Adv. Math. Commun. 3 (1): 59-81, 2009].

The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ and odd q. In particular, the relevant family contains at least $\begin{cases} \left\lfloor\frac{q^t+1}{8rt}\right\rfloor,& \text{ if }t\not\equiv 2\pmod{4};\\[8pt] \left\lfloor\frac{q^t+1}{4rt(q^2+1)}\right\rfloor,& \text{ if }t\equiv 2\pmod{4}, \end{cases}$ inequivalent members for given $q=p^r$ and $n=2t>8$, where $p=\mathrm{char}(\mathbb{F}_q)$. This is a great improvement of previous results: for given q and $n>8$, the number of inequivalent maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ in all classes known so far, is smaller than $q^2$. Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil$ inequivalent APN functions on the finite field with ${2^{2m}}$ elements, where $\varphi$ denotes Euler’s totient function. This is a great improvement of previous results: for even m, the best known lower bound has been $\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor +1\right)$; for odd m, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.

Let n be a positive integer and I a k-subset of integers in [0,n−1]. Given a k-tuple A=(α0,⋯,αk−1)∈Fqnk, let MA,I denote the matrix (αiqj) with 0≤i≤k−1 and j∈I. When I={0,1,⋯,k−1}, MA,I is called a Moore matrix which was introduced by E. H. Moore in 1896. It is well known that the determinant of a Moore matrix equals 0 if and only if α0,⋯,αk−1 are Fq-linearly dependent. We call I that satisfies this property a Moore exponent set. In fact, Moore exponent sets are equivalent to maximum rank-distance (MRD) code with maximum left and right idealizers over finite fields. It is already known that I={0,⋯,k−1} is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered in [5] both give rise to new Moore exponent sets. By using algebraic geometry approach, we obtain an asymptotic classification result: for q>5, if I is not an arithmetic progression, then there exists an integer N depending on I such that I is not a Moore exponent set provided that n>N.

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in Fqn×n the idealizers have been proved to be isomorphic to finite fields of size at most qn. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in Fqn×n for n≤9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for n=7, q odd and for n=8, q≡1(mod3). These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n≥9.

Let d,n∈Z+ such that 1≤d≤n. A d-code C⊂Fqn×n is a subset of order n square matrices with the property that for all pairs of distinct elements in C, the rank of their difference is greater than or equal to d. A d-code with as many as possible elements is called a maximum d-code. The integer d is also called the minimum distance of the code. When d<n, a classical example of such an object is the so-called generalized Gabidulin code (Gabidulin and Kshevetskiy (2005)). In Delsarte and Goethals (1975), Schmidt (2015) and K.U. Schmidt (2018), several classes of maximum d-codes made up respectively of symmetric, alternating and Hermitian matrices were exhibited. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric 2-code which is not equivalent to the one with same parameters constructed in Schmidt (2015).