Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let \(\alpha \) be a generator of \(\mathbb F_{3^m}\setminus \{0\}\), where m is a positive integer. Denote by \(\mathcal {C}_{(i_1,i_2,\cdots , i_t)}\) the cyclic code with generator polynomial \(m_{\alpha

In this paper, we promote Trojan message attacks against Merkle–Damgård hash functions and their concatenation combiner in quantum settings for the first time. Two main quantum scenarios are considered, involving the scenarios where a substantial amount of cheap quantum random access memory (qRAM) is available and where qRAM is limited and expensive to access. We first discuss the construction of diamond

Combinatorial neural (CN) codes are binary codes introduced firstly by Curto et al. for asymmetric channel, and then are further studied by Cotardo and Ravagnani under the metric \(\delta _r\) (called asymmetric discrepancy) which measures the differentiation of codewords in CN codes. When \(r>1\), CN codes are different from the usual error-correcting codes in symmetric channel (\(r=1\)). In this

We study the complexity of the Code Equivalence Problem on linear error-correcting codes by relating its variants to isomorphism problems on other discrete structures—graphs, lattices, and matroids. Our main results are a fine-grained reduction from the Graph Isomorphism Problem to the Linear Code Equivalence Problem over any field \(\mathbb {F}\), and a reduction from the Linear Code Equivalence Problem

Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field \(\mathbb {F}_q\). We consider the relationship between projective Reed–Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when q is sufficiently large, the dimension of the hull

Using codes defined over \(\mathbb {F}_4\) and \(\mathbb {F}_2 \times \mathbb {F}_2\), we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields \(\mathbb {Q}(\sqrt{d})\) with \(d \equiv 1\mod 4\) a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend

Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding

A bijection \(\theta :G\rightarrow G\) of a finite group G is an orthomorphism of G if the mapping \(x\mapsto x^{-1}\theta (x)\) is also a bijection. Two orthomorphisms \(\theta \) and \(\phi \) of a finite group G are orthogonal if the mapping \(x\mapsto \theta (x)^{-1}\phi (x)\) is also bijective. We show that there is a pair of orthogonal orthomorphisms of a finite nilpotent group G if and only

Brawley and Carlitz introduced diamond products of elements of finite fields and associated composed products of polynomials in 1987. Composed products yield a method to construct irreducible polynomials of large composite degrees from irreducible polynomials of lower degrees. We show that the composed product of two irreducible polynomials of degrees m and n is again irreducible if and only if m and

In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order 64. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces

In this work we present results on the classification of \(\mathbb {F}_{q^n}\)-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes \(\mathcal {C}=\langle X^{q^t}, F(X), G(X) \rangle \subseteq \mathcal {L}_{n,q}\) of exceptional type, i.e. such that \(\mathcal {C}\) is MRD over infinitely

Set systems with strongly restricted intersections, called \(\alpha \)-intersecting families for a vector \(\alpha \), were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector \(\alpha =(a_1, \ldots , a_k)\), a collection \({\mathcal {F}}\) of subsets over an n element set is an \(\alpha \)-intersecting

In recent years, it has become a popular trend to propose quantum versions of classical attacks. The rectangle attack as a differential attack is widely used in symmetric cryptanalysis and applied on many block ciphers. To improve its efficiency, we propose a new quantum rectangle attack firstly. In rectangle attack, it counts the number of valid quartets for each guessed subkeys and filters out subkey

Achieving tight security is a fundamental task in cryptography. While one of the most important purposes of this task is to improve the overall efficiency of a construction (by allowing smaller security parameters), many current lattice-based instantiations do not completely achieve the goal. Particularly, a super-polynomial modulus seems to be necessary in all prior work for (almost) tight schemes

The Brakerski–Gentry–Vaikuntanathan (BGV) scheme is a Fully Homomorphic Encryption (FHE) cryptosystem based on the Ring Learning With Error (RLWE) problem. Ciphertexts in this scheme contain an error term that grows with operations and causes decryption failure when it surpasses a certain threshold. Consequently, the parameters of BGV need to be estimated carefully, with a trade-off between security

It is important to study the new construction methods of bent functions. In this paper, we first propose a secondary construction method of \((k+s)\)-variable bent function g through a family of s-plateaued functions \(f_0,f_1,\ldots ,f_{2^s-1}\) on k variables with disjoint Walsh supports, which can be obtained through any given \((k-s)\)-variable bent function f by selecting \(2^s\) disjoint affine

Let n be an odd positive integer, p be an odd prime with \(p\equiv 3\pmod 4\), \(d_{1} = {{p^{n}-1}\over {2}} -1 \) and \(d_{2} =p^{n}-2\). The function defined by \(f_u(x)=ux^{d_{1}}+x^{d_{2}}\) is called the generalized Ness–Helleseth function over \(\mathbb {F}_{p^n}\), where \(u\in \mathbb {F}_{p^n}\). It was initially studied by Ness and Helleseth in the ternary case. In this paper, for \(p^n

The community has been pursuing improvements in the cardinalities for constant dimensional codes (CDC for short) for the past decade. Lao et al. (IEEE Trans Inf Theory 69(7):4333–4344, 2023) has shown that mixed dimension subspace codes can be used to construct large constant dimension subspace codes. The exploration of the CDCs’ construction is transformed into finding mixed dimension/distance subspace

Constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. This paper introduces a new class of codes, namely generalized bilateral Ferrers diagram rank-metric codes, to generalize the bilateral multilevel construction in Etzion and Vardy (Adv Math Commun 16:1165–1183, 2022). Combining our generalized bilateral multilevel construction and the

In this article we prove Griesmer type bounds for additive codes over finite fields. These new bounds give upper bounds on the length of maximum distance separable (MDS) codes, codes which attain the Singleton bound. We will also consider codes to be MDS if they attain the fractional Singleton bound, due to Huffman. We prove that this bound in the fractional case can be obtained by codes whose length

In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in \({{\,\textrm{PG}\,}}(n,q)\) with q odd. We define relations on the anisotropic points of such a quadric that depend on the type of line spanned by the points and whether or not they are of the same “quadratic type”. This yields

In the combinatorial context, one of the key problems in sequence reconstruction is to find the largest intersection of two metric balls of radius r. In this paper we study this problem for permutations of length n distorted by Hamming errors and determine the size of the largest intersection of two metric balls with radius r whose centers are at distance \(d=2,3,4\). Moreover, it is shown that for

Secret sharing is a general method for distributing sensitive data among the participants of a system such that only a collection of predefined qualified coalitions can recover the secret data. One of the most widely used special cases is threshold secret sharing, where every subset of participants of size above a given number is qualified. In this short note, we propose a general construction for

For a q-polynomial L over a finite field \(\mathbb {F}_{q^n}\), we characterize the differential spectrum of the function \(f_L:\mathbb {F}_{q^n} \rightarrow \mathbb {F}_{q^n}, x \mapsto x \cdot L(x)\) and show that, for \(n \le 5\), it is completely determined by the image of the rational function \(r_L :\mathbb {F}_{q^n}^* \rightarrow \mathbb {F}_{q^n}, x \mapsto L(x)/x\). This result follows from

This paper focuses on non-existence results for Cameron–Liebler k-sets. A Cameron–Liebler k-set is a collection of k-spaces in \({{\,\mathrm{\textrm{PG}}\,}}(n,q)\) or \({{\,\mathrm{\textrm{AG}}\,}}(n,q)\) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron–Liebler k-sets with parameter x. This

Tang and Ding (IEEE Trans Inf Theory 67(1):244–254, 2021) opened a new direction of searching for t-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over \(\textrm{GF}(3^{m})\). Based on them, we present more

This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let \(\mathbb {F}_q\) denote the finite field with q elements, and let G be a finite Abelian group of order n. The

Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in Özbudak and Pelen (J Algebr Comb 56:635–658, 2022), Özbudak and Pelen constructed infinite families of symmetric association schemes of

Given a polynomial f over the finite field \(\mathbb {F}_q\), its intersection distribution provides fruitful information on how lines in the affine plane intersect the graph of f over \(\mathbb {F}_q\). The intersection distribution in its simplest cases gives rise to oval polynomials in finite geometry and Steiner triple systems in design theory. Previously, the intersection distribution of degree

It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one \(-1\) entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. The proofs rely on solving polynomial systems of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard

For each odd prime power q, we present two rational functions \(f(X)\in \mathbb {F}_q(X)\) which have the unusual property that, for every odd n, the function induced by f(X) on \(\mathbb {F}_{q^n}\setminus \mathbb {F}_q\) is \((q-1)\)-to-1.

By using the notion of a d-embedding \(\Gamma \) of a (canonical) subgeometry \(\Sigma \) and of exterior sets with respect to the h-secant variety \(\Omega _{h}({\mathcal {A}})\) of a subset \({\mathcal {A}}\), \( 0 \le h \le n-1\), in the finite projective space \({\textrm{PG}}(n-1,q^n)\), \(n \ge 3\), in this article we construct a class of non-linear (n, n, q; d)-MRD codes for any \( 2 \le d \le

Recently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on the prime field \({\mathbb {F}}_p\) for some large prime p. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives

Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural q-analog of a subcube partition of the hypercube (which can be seen as a polar space with \(q=1\)). Our

Complex orthogonal designs (CODs) have been used to construct space-time block codes. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of balanced complex orthogonal designs (BCODs). The code

A (v, k, t) packing of size b is a system of b subsets (blocks) of a v-element underlying set such that each block has k elements and every t-set is contained in at most one block. P(v, k, t) stands for the maximum possible b. A packing is called abundant if \(b> v\). We give new estimates for P(v, k, t) around the critical range, slightly improving the Johnson bound and asymptotically determine the

Proxy re-encryption (PRE) is a cryptosystem that realizes efficient encrypted data sharing by allowing a third party proxy to transform a ciphertext intended for a delegator (i.e., Alice) to a ciphertext intended for a delegatee (i.e., Bob). Attribute-based proxy re-encrypftion (AB-PRE) generalizes PRE to the attribute-based scenarios, enabling fine-grained access control on ciphertexts. However, the

Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor

For an odd prime power q, let \(\mathbb {F}_{q^2}=\mathbb {F}_q(\alpha )\), \(\alpha ^2=t\in \mathbb {F}_q\) be the quadratic extension of the finite field \(\mathbb {F}_q\). In this paper, we consider the irreducible polynomials \(F(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-\cdots -c_{2}^qx^2+c_{1}^qx-1\) over \(\mathbb {F}_{q^2}\), where k is an odd integer and the coefficients \(c_i\) are in the form \(c_i=a_i+b_i\alpha