Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. The objective of this paper is to study how to obtain $3$-designs with $2$-transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are $2$-transitive, is considered. A characterization of such incidence structure to be a $3$-design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of $3$-designs are constructed by employing APN functions. Some $3$-designs presented in this paper give rise to self-dual binary codes or linear codes with optimal or best parameters known. Several conjectures on $3$-designs and binary codes are also presented.
In this work, we present two new families of quadratic APN functions. The first one (F1) is constructed via biprojective polynomials. This family includes one of the two APN families introduced by Göloǧlu in 2022. Then, following a similar approach as in Li \emph{et al.} (2022), we give another family (F2) obtained by adding certain terms to F1. As a byproduct, this second family includes one of the two families introduced by Li \emph{et al.} (2022). Moreover, we show that for $n=12$, from our constructions, we can obtain APN functions that are CCZ-inequivalent to any other known APN function …
In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over $\gf_{2^{2m}}$ with $m$ even, i.e., $\mathcal{F}_1(x)=x^{j(2^m-1)}$ with $\gcd(j,2^m+1)=1$ and $\mathcal{F}_2(x)=x^{j(2^m-1)+1}$ with $j = \frac{2^m+2}{3}$. As far as the authors know, our infini…
The investigation of partially APN functions has attracted a lot of research interest recently. In this paper, we present several new infinite classes of 0-APN power functions over $\mathbb{F}_{2^n}$ by using the multivariate method and resultant elimination, and show that these 0-APN power functions are CCZ-inequivalent to the known ones.
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{ö}lo{ǧ}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 $Γ$-ranks of our APN classes over $\gf_{2^8}$ or $\gf_{2^…
Partially APN functions attract researchers' particular interest recently. It plays an important role in studying APN functions. In this paper, based on the multivariate method and resultant elimination, we propose several new infinite classes of $0$-APN power functions over $\mathbb{F}_{2^n}$. Furthermore, two infinite classes of $0$-APN power functions $x^d$ over $\mathbb{F}_{2^n}$ are characterized completely where $(2^k-1)d\equiv 2^m-1~({\rm mod}\ 2^n-1)$ or $(2^k+1)d\equiv 2^m+1~({\rm mod}\ 2^n-1)$ for some positive integers $n, m, k$. These infinite classes of $0$-APN power functions can…
In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field $\F_2$, extending some earlier results of Leander and Rodier.
Recently, the investigation of Partially APN functions has attracted a lot of attention. In this paper, with the help of resultant elimination and MAGMA, we propose several new infinite classes of 0-APN power functions over $\mathbb{F}_{2^{n}}$. By the main result in [4], these $0$-APN power functions are CCZ-inequivalent to the known ones. Moreover, these infinite classes of 0-APN power functions can explain some exponents for $1\leq n\leq11$ which are not yet ``explained" in the tables of Budaghyan et al. [3].
Linear codes generated by component functions of perfect nonlinear (PN) and almost perfect nonlinear (APN) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$-ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$-ary codes are flexible. Many of the codes presented in …
In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differential attack (or their variants). Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd(2^m+1,2^k+1)=1$, we investigate the locally-APN-ness property of an infinite family of Niho type power functions in the form $F(x)=x^{s(2^m…
In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the form $x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,ω, ω^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,ω, ω^2,1)$ in which $ω\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking $a=ω$ and $b=c=ω^2$, we observe that such kind of quadrinomials can be rewritten a…
Dillon observed that an APN function $F$ over $\mathbb{F}_2^{n}$ with $n$ greater than $2$ must satisfy the condition $\{F(x) + F(y) + F(z) + F(x + y + z) \,:\, x,y,z \in\mathbb{F}_2^n\}= \mathbb{F}_2^n$. Recently, Taniguchi (2023) generalized this condition to functions defined from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m>n$, calling it the D-property. Taniguchi gave some characterizations of APN functions satisfying the D-property and provided some families of APN functions from $\mathbb{F}_2^n$ to $\mathbb{F}_2^{n+1}$ satisfying this property. In this work, we further study the D-prop…
In this paper, we study the boomerang spectrum of the power mapping $F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$ is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from $(p,k)=(2,1)$ to general $(p,k)$. We then determine the boomerang spectrum of $F(x)$ by making use of its differential spectrum, which shows that the boomerang uniformity of $F(x)$ is 4 if $p=2$ and $m$ is odd and otherwise it is 2. Our re…
Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional APN if it is APN on infinitely many extensions of $\mathbb{F}_{2^n}$. Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number $(2^k+1)$ or a Kasami…
Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permut…
Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McG…
In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $…
Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over $\gf(q)…
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming an upper bound on the order of the monomial automorphism groups, according to computational results.
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters $[n, k]$ and $(n-6)/3 \leq k \leq 2(n+6)/3$. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and …
An $[n, k, n-k+1]$ linear code is called an MDS code. An $[n, k, n-k]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. This ternary code holds $4$-designs, and its extended code holds a Steiner system $S(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding $t$-d…
Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with $(r,δ)$-locality that achieve the equality in the Singleton-type bound. We use matrix-product codes to propose two infinite families of $q$-ary optimal $(r,δ)$ locally repairable codes of lengths up to $q^2+q$. The ingredients in the matrix-product codes are either linear maximum distance separable codes or optimal locally repairable codes of small lengths. Further analysis and refinement yield a construction of another infini…
The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the largest $t$ for which an infinite family of $t$-designs is derived directly from a linear or nonlinear code is $t=3$. Sporadic $4$-designs and $5$-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of $2$-designs and $3$-designs from linear codes. The parameters of some known $t$-designs are also derived. In addition, many conjectured infinite families of $2$-…
Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance t…
It has been known for a long time that $t$-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a $t$-design. While a lot of progress in the direction of constructing codes from $t$-designs has been made, only a small amount of work on the construction of $t$-designs from codes has been done. The objective of this paper is to construct infinite families of $2$-designs and $3$-designs from a type of binary linear codes with five-weights. The total number of $2$-designs and $3$-designs obtained in this paper are exponent…
Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $t$-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, by determining the weight distribution of a class of linear codes derived from…
Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have been attracted lots of research interest for decades. The interplay between coding theory and $t$-designs has on going for many years. As we all known, $t$-designs can be used to derive linear codes over any finite field, as well as the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then…
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $\mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these codes support infinite families of 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $\mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m \geqslant 5$ is odd and $…
We present new constructions for perfect and odd perfect sequences over the quaternion group $Q_8$. In particular, we show for the first time that perfect and odd perfect quaternion sequences exist in all lengths $2^t$ for $t\geq0$. In doing so we disprove the quaternionic form of Mow's conjecture that the longest perfect $Q_8$-sequence that can be constructed from an orthogonal array construction is of length 64. Furthermore, we use a connection to combinatorial design theory to prove the existence of a new infinite class of Williamson sequences, showing that Williamson sequences of length $2…
