The mappingχnfrom Fn2to itself defined byy=χn(x) withyi=xi+xi+2(1 +xi+1), where the indices are computed modulo n, has been widely studied for its applications in lightweight cryptography. However,χnis bijective on Fn2only whennis odd, restricting its use to odd-dimensional vector spaces over F2. To address this limitation, we introduce and analyze the generalized mappingχn,mdefined byy=χn,m(x) withyi=xi+xi+m(xi+m−1+ 1)(xi+m−2+ 1) · · · (xi+1+ 1), wheremis a fixed integer withm∤n. To investigate such mappings, we further generalizeχn,mto θm,k, where θm,kis given byyi=xi+mkПmk−1j=1, m∤j(xi+j+ 1) , fori∈ {0, 1, . . . ,n− 1}. We prove that these mappings generate an abelian group isomorphic to the group of units in F2[z]/(z⌊n/m⌋+1). This structural insight enables us to construct a broad class of permutations over Fn2for any positive integern, along with their inverses. We rigorously analyze algebraic properties of these mappings, including their iterations, fixed points, and cycle structures. Additionally, we provide a comprehensive database of the cryptographic properties for iterates ofχn,mfor small values ofnandm. Finally, we conduct a comparative security and implementation cost analysis amongχn,m,χn,χχn(EUROCRYPT 2025 [3]) and their variants, and prove Conjecture 1 proposed in [3] as a by-product of our study. Our results lead to generalizations ofχn, providing alternatives toχnandχχn.

The minimum storage rack-aware regenerating (MSRR) code is a variation of regenerating codes that achieves the optimal repair bandwidth for a single node failure within the rack-aware model. We study the access complexity of repairing MSRR codes, that allows collective information processing among nodes within the same rack. A previous study has reported construction of MSRR codes that require accessing the minimum number of symbols to repair a single node. We extend this work by constructing a family of MSRR codes that minimizes the number of symbols accessed to repair sequentially ordered failed nodes in a single rack, and further show that for certain code parameters, another version of MSRR codes can be constructed with reduced sub-packetization while still preserving the optimal access property.

Motivated by high-density storage needs, symbol-pair codes were introduced by Cassuto and Blaum to address channels with overlapping symbol outputs. In this paper, we present a systematic study of two families of reducible cyclic codes under the symbol-pair metric. By employing analytical techniques rooted in cyclotomic numbers and Gaussian period theory over finite fields, we characterize the admissible symbol-pair weights of these codes. Significantly, we demonstrate that their minimum symbol-pair distances attain twice the minimum Hamming distances under specific algebraic constraints. Furthermore, we identify and rigorously determine the symbol-pair weight distributions for several three-weight code families. Notably, we construct a class of MDS symbol-pair codes that achieve optimal distance parameters by the puncturing technique. As supplementary contributions, the paper resolves several computational problems concerning generalized cyclotomic numbers, thereby enriching the mathematical foundation for code parameter analysis.

BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of BCH codes has been made, but little is known about the properties of their duals. Recently, in order to study the duals of BCH codes and the lower bounds on their minimum distances, a new concept called dually-BCH code was proposed by (Gong et al., 2022). In this paper, the lower bounds on the minimum distances of the duals of narrow-sense BCH codes with length$\frac {q^{m}-1}{\lambda }$over$\mathbb {F}_{q}$are developed, where$\lambda $is a positive integer satisfying$\lambda =q^{s}-1$and$s\, |\,m$, or$\lambda \, |\, q-1$. In addition, the sufficient and necessary conditions in terms of the designed distances for these codes being dually-BCH codes are presented. Our lower bounds on the minimum distances of the duals of BCH codes include the bounds stated in (Gong et al., 2022) as a special case. Moreover, our lower bounds improve the bounds stated in (Gong et al., 2022), the classical Sidel’nikov bound, and the Carlitz-Uchiyama bound when the designed distances of the BCH codes are in some ranges. Several examples show that our proposed lower bounds are good in some cases.

Let${\mathcal C}$be an$[n, k]$linear code over the finite field${\mathbb F}_{q}$. Let$d_{I}({\mathcal C})$denote its insertion-deletion (insdel for short) distance, which characterizes the insdel error-correcting capability of${\mathcal C}$. To determine the insdel distances of linear codes is a very challenging problem. In this paper we propose a strict half-Singleton upper bound$d_{I}({\mathcal C}) \leq 2(n-2k+1)$if${\mathcal C}$does not contain the codeword with all 1s, which generalizes the half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li, and a stronger direct upper bound$d_{I}({\mathcal C}) \leq 2(d_{H}({\mathcal C})-t)$under a weak condition, where$t\geq 1$is a positive integer determined by the generator matrix and$d_{H}({\mathcal C})$denotes the Hamming distance of${\mathcal C}$. A sufficient condition for a linear code attaining the strict half-Singleton bound is given. We prove that the code length of an optimal binary linear insdel code with respect to the (strict) half-Singleton bound is about twice its dimension and conjecture that optimal binary linear insdel codes have exact parameters$[{2k, k, 4}]$or$[{2k+1, k, 4}]$with respect to the half-Singleton bound or the strict half-Singleton bound, respectively. Moreover, interestingly explicit optimal linear insdel codes attaining the (strict) half-Singleton bound, with the code length being independent of the finite field size, are given.

The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (2021) and Li and Li (2022) obtained the complete weight hierarchy of linear codes from quadratic forms over finite fields of odd characteristic by analysis of the solutions of the restricted quadratic equation in its subspace. In this paper, we further determine the complete weight hierarchy of linear codes from quadratic forms over finite fields of even characteristic by carefully studying the behavior of the corresponding restricted quadratic forms to the subspaces of the field, and complement the results of Li and Li.

Permutation polynomials have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. An important subfamily of permutations is the class of involutions (those permutations are equal to their compositional inverse). Elements of this class have been used frequently for block cipher designs and coding theory. In this article, we further investigate this corpus using new approaches, specifically from two-to-one (2-to-1) functions and (in some cases) using the graph indicators introduced by Carlet in 2020. In our constructions of involutions over the finite field$\mathbb {F}_{2^{n}}$of order$2^{n}$, we shall intensively use 2-to-1 mappings over$\mathbb {F}_{2^{n}}$. More specifically, we present a new constructive method to design involutions from 2-to-1 mappings through their graph indicator and derive new involutions from known 2-to-1 mappings. Besides, we also propose several new classes of 2-to-1 mappings, including 2-to-1 hexanomials, 2-to-1 mappings of the form$(x^{2^{k}}+x+\delta)^{s_{1}}+(x^{2^{k}}+x+\delta)^{s_{2}}+cx$, and 2-to-1 mappings from linear 2-to-1 mappings. We also exhibit the corresponding involutions of the constructed 2-to-1 mappings. Furthermore, an infinite family of involutions with differential uniformity at most 4 (EA-inequivalent to the inverse function) is obtained. Finally, we highlight that all our derived families of involutions have no fixed point, further accentuating their cryptographic interest.

The minimum storage rack-aware regenerating (MSRR) code is a variation of regenerating codes that achieves the optimal repair bandwidth for a single node failure in the rack-aware model. Some explicit constructions of MSRR codes for all parameters to repair a single failed node have been reported. This paper studies MSRR codes with error-correcting capability for multiple erasure tolerance. First, we propose a general repair model of maximum distance separable (MDS) codes with error-correcting capability for multiple erasure tolerance and derive lower bounds on the number of symbols downloaded and accessed, respectively from helper racks for the purpose of correction and repair. Then, we construct a class of MDS array codes and scalar Reed-Solomon (RS) codes with the optimal repair bandwidth and error resilient capability for multiple node failures. Further, our codes are shown to have the low-access property. In particular, they have the optimal access property for repairing$u$failed nodes when the dimension of the code is divisible by the rack size$u$.

An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. [5]. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2] generalized the construction of cyclic r-LRCs proposed by Tamo et al. [19], [20] and constructed several classes of optimal (r, δ)-LRCs of length n for n (q-1) or n (q+1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of [2], [3], this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q+1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r, δ)-LRCs proposed in [2], [3], and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q + 1), respectively with (r + δ - 1) n can be obtained from our method.

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f)={ Tr(af(x)+bx)x ∈ \mathbb Fqm*: a,b ∈ \mathbb Fqm }, where q is a prime power, \mathbb Fqm* = \mathbb Fqm \{0}, Tr is the trace function from \mathbb Fqm to \mathbb Fq, and f(x) is a function from \mathbb Fqm to \mathbb Fqm with f(0)=0. Almost bent functions, quadratic functions and some monomials on \mathbb F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for$x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$being involutions over the finite field${\mathbb F}_{q}$, where$r\geq 1$and$s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$from given involutions over some subgroups of${\mathbb F}_{q}^{*}$by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$are obtained.

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules. The hyperexponential solutions are computed recursively by solving a sequence of first-order ordinary matrix equations. As the recursion proceeds, the matrix equations will have constant coefficients with respect to the operators that have been considered.