We initiate the study of λ-fold near-factorizations of groups with λ>1. While λ-fold near-factorizations of groups with λ=1 have been studied in numerous papers, this is the first detailed treatment for λ>1. We establish fundamental properties of λ-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of λ-fold near-factorizations, including upper bounds on λ. We present three constructions of infinite families of λ-fold near-factorizations, highlighting the characterization of two subfamilies of λ-fold near-factorizations. We discuss a computational approach to λ-fold near-factorizations and tabulate computational results for abelian groups of small order.

We compute the weight distribution of binary cyclic codes with arbitrarily many generalized Niho-type nonzeroes, where the exponents of the nonzeroes nearly form a long arithmetic progression. Determining their weight distribution requires two distinct techniques. The first involves using the polar representation associated with generalized Niho-type exponents and leveraging the subtle structure of the nonzeroes whose exponents are close to an arithmetic progression. The second applies the weight distribution of binary Zetterberg codes to address certain counting problems. This resolves two open questions raised by Li and Zeng.

Abstract Given a polynomial f over the finite field $$\mathbb {F}_q$$ F q , its intersection distribution provides fruitful information on how lines in the affine plane intersect the graph of f over $$\mathbb {F}_q$$ 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 two and degree three polynomials has been computed. In this paper, we determine the intersection distribution of all degree four polynomials over finite fields. As an application, we present a direct construction of Steiner systems using polynomials with prescribed intersection distribution.

This article offers an integrated solution for first-person shooter (FPS) games to train agents with adaptive strategies. Solving such complex decision tasks requires generalization ability and adaptive strategies. We develop a framework using a novel adaptive strategic control algorithm combined with advanced techniques, such as the hindsight experience replay, multiagent reinforcement learning, and league training. The approach adopts a multistage learning scheme, consisting of learning a goal-conditioned navigation policy, then transferring to learn sophisticated shooting skills by playing against a league of players, and finally learning adaptive strategies. Our agent achieves the SOTA result in pastViZDoomAI Competitions, surpassing previous top-ranked agents (never seen during training) by a large margin. We provide comprehensive analysis and experiments to elaborate the effect of each component in affecting the agent performance and demonstrate that the proposed and adopted techniques are essential to achieve superior performance inViZDoomCompetition and potentially valuable for general end-to-end FPS games.

Goal-conditioned reinforcement learning (RL) with sparse rewards remains a challenging problem in deep RL. Hindsight Experience Replay (HER) has been demonstrated to be an effective solution, where HER replaces desired goals in failed experiences with practically achieved states. Existing approaches mainly focus on either exploration or exploitation to improve the performance of HER. From a joint perspective, exploiting specific past experiences can also implicitly drive exploration. Therefore, we concentrate on prioritizing both original and relabeled samples for efficient goal-conditioned RL. To achieve this, we propose a novel value consistency prioritization (VCP) method, where the priority of samples is determined by the consistency of ensemble Q-values. This distinguishes the VCP method with most existing prioritization approaches which prioritizes samples based on the uncertainty of ensemble Q-values. Through extensive experiments, we demonstrate that VCP achieves significantly higher sample efficiency than existing algorithms on a range of challenging goal-conditioned manipulation tasks. We also visualize how VCP prioritizes good experiences to enhance policy learning.

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$ . We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well studied subclass of BCH codes, there are very few theoretical results on the minimum distance. In this paper, we present new results on the minimum distance of narrow-sense primitive BCH codes with special Bose distance. We prove that for a prime power $q$, the $q$-ary narrow-sense primitive BCH code with length $q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \le m-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$. This is achieved by employing the beautiful theory of sets of quadratic forms, symmetric bilinear forms, and alternating bilinear forms over finite fields, which can be best described using the framework of association schemes.

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over GF(q) with length n = (qm-1)/(q -1). Little is known about this class of BCH codes when q > 2. The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length n = (3m- 1)/2 are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros.

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters.

Compressed sensing is a novel sampling theory, which provides a fundamentally new approach to data acquisition. It asserts that a sparse or compressible signal can be reconstructed from much fewer measurements than traditional methods. A central problem in compressed sensing is the construction of the sensing matrix. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Among them, most constructions are based on coherence, which essentially generates matrices with low coherence. In this paper, we introduce the concept of near orthogonal systems to characterize the matrices with low coherence, which lie in the heart of many different applications. The constructions of these near orthogonal systems lead to deterministic constructions of sensing matrices. We obtain a series of m×n binary sensing matrices with sparsity level k=Θ(m(1/2)) or k=O((m/logm)(1/2)). In particular, some of our constructions are the best possible deterministic ones based on coherence. We conduct a lot of numerical experiments to show that our matrices arising from near orthogonal systems outperform several typical known sensing matrices.

The determination of the cross correlation between an m-sequence and its decimated sequence has been a longstanding research problem. Considering a ternary m-sequence of period 33r- 1, we determine the cross correlation distribution for decimations d = 3r+ 2 and d = 32r+ 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 22lm- 1, we make an initial investigation for the decimation d = (22lm- 1)/(2m+ 1) + 2s, where l ≥ 2 is even and 0 <; s <; 2m - 1. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.

The determination of weight distribution of cyclic codes involves the evaluation of Gauss sums and exponential sums. Despite some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectra of Hermitian forms graphs.

Compressed sensing is a sampling technique which provides a fundamentally new approach to data acquisition. Comparing with traditional methods, compressed sensing makes full use of sparsity so that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Inspired by algebraic geometry codes, we introduce a new deterministic construction via algebraic curves over finite fields, which is a natural generalization of DeVore's construction using polynomials over finite fields. The diversity of algebraic curves provides numerous choices for sensing matrices. By choosing appropriate curves, we are able to construct binary sensing matrices which are superior to Devore's ones. We hope this connection between algebraic geometry and compressed sensing will provide a new point of view and stimulate further research in both areas.