arXiv subjects

Fang-Wei Fu

Publications and source records attributed to Fang-Wei Fu.

At least 37 records · Page 2Linked to original sources

arXiv2026

A class of pseudorandom sequences From Function Fields

Motivated by the constructions of pseudorandom sequences over the cyclic elliptic function fields by Hu \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 53(7), 2007]} and the constructions of low-correlation, large linear span binary sequences from function fields by Xing \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]}, we utilize the bound derived by Weil \text{[Basic Number Theory, Grund. der Math. Wiss., Bd 144]} and Deligne \text{[ Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977)]} for the exponential sums over the general algebraic function fields and study the periods, linear complexities, linear complexity profiles, distributions of $r-$patterns, period correlation and nonlinear complexities for a class of $p-$ary sequences that generalize the constructions in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]} and [IEEE Trans. Inf. Theory, 53(7), 2007].

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arXiv2026

Extended Gabidulin-Kronecker Product Codes and Their Application to Cryptosystems

In this paper, we initiate the study of Extended Gabidulin codes with a Kronecker product structure and propose three enhanced variants of the Rank Quasi-Cyclic (RQC) (Melchor et.al., IEEE IT, 2018) cryptosystem. First, we establish precise bounds on the minimum rank distance of Gabidulin-Kronecker product codes under two distinct parameter regimes. Specifically, when $n_{1}=k_{1}$ and $n_{2}=m<n_{1}n_{2}$, the minimum rank distance is exactly $n_{2}-k_{2}+1$. This yields a new family of Maximum Rank Distance (MRD) codes, which are distinct from classical Gabidulin codes. For the case of $k_{1}\leq n_{1},k_{2}\leq n_{2},n_{1}n_{2}\leq m$, the minimum rank distance $d$ of Gabidulin-Kronecker product codes satisfies a tight upper and lower bound, i.e., $n_{2}-k_{2}+1 \leq d \leq (n_{1}-k_{1}+1)(n_{2}-k_{2}+1)$. Second, we introduce a new class of decodable rank-metric codes, namely Extended Gabidulin-Kronecker product (EGK) codes, which generalize the structure of Gabidulin-Kronecker product (GK) codes. We also propose a decoding algorithm that directly retrieves the codeword without recovering the error vector, thus improving efficiency. This algorithm achieves zero decoding failure probability when the error weight is within its correction capability. Third, we propose three enhanced variants of the RQC cryptosystem based on EGK codes, each offering a distinct trade-off between security and efficiency. For 128-bit security, all variants achieve significant reductions in public key size compared to the Multi-UR-AG (Bidoux et.al., IEEE IT, 2024) while ensuring zero decryption failure probability--a key security advantage over many existing rank-based schemes.

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arXiv2025

Sequence Reconstruction Problem for Ternary Deletion Channels

The sequence reconstruction problem was proposed by Levenshtein in 2001. In this model, a sequence from a code is transmitted over several channels, and the decoder receives the distinct outputs from each channel. The main problem is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the sequence reconstruction problem is equivalent to finding the value of $N_q(n,d,t)$, defined as the size of the largest intersection of two metric balls of radius $t$, where the distance between their centers is at least $d$ and the sequences are $q$-ary sequences of the length $n$. Levenshtein first discussed this problem in the uncoded sequence setting and determined the value of $N_q(n,1,t)$ for any $n\geqslant t$. Moreover, Gabrys and Yaakobi studied this problem in the context of binary one-deletion-correcting codes and determined the value of $N_2(n,2,t)$ for $t\geqslant 2$. In this paper we study this problem for $3$-ary sequences of length $n$ over the deletion channel, where the transmitted sequence belongs to a one-deletion-correcting code and there are $t$ deletions in every channel. Specifically, we determine $N_3(n,2,t)$ for $t\geqslant 2$.

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arXiv2025

Further results on bent partitions

Bent partitions of $V_{n}^{(p)}$ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the finite field $\mathbb{F}_{p}$, $n$ is an even positive integer, and $p$ is a prime. For bent partitions, there remains a challenging open problem: Whether the depth of any bent partition of $V_{n}^{(p)}$ is always a power of $p$. Notably, the depths of all current known bent partitions of $V_{n}^{(p)}$ are powers of $p$. In this paper, we prove that for a bent partition $Γ$ of $V_{n}^{(p)}$ for which all the $p$-ary bent functions generated by $Γ$ are regular or all are weakly regular but not regular, the depth of $Γ$ must be a power of $p$. We present new constructions of bent partitions that (do not) correspond to vectorial dual-bent functions. In particular, a new construction of vectorial dual-bent functions is provided. Additionally, for general bent partitions of $V_{n}^{(2)}$, we establish a characterization in terms of Hadamard matrices.

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arXiv2025

Three Classes of Twisted Gabidulin Codes with Different Twists

Twisted Gabidulin codes are an extension of Gabidulin codes and have recently attracted great attention. In this paper, we study three classes of twisted Gabidulin codes with different twists. Moreover, we establish necessary and sufficient conditions for them to be maximum rank distance (MRD) codes, determine the conditions under which they are not MRD codes, and construct several classes of MRD codes via twisted Gabidulin codes. In addition, considering these codes in the Hamming metric, we provide necessary and sufficient conditions for them to be maximum distance separable (MDS), almost MDS, or near MDS. Finally, we investigate the covering radii and deep holes of twisted Gabidulin codes.

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arXiv2025

Weight Distribution of Repeated-Root Cyclic Codes with Prime Power Lengths

Determining the weight distribution of a linear code is a classical and fundamental topic in coding theory that has been extensively investigated. Repeated-root cyclic codes, which form a significant subclass of error-correcting codes, have found broad applications in quantum error-correcting codes, symbol-pair codes, and storage codes. Through polynomial derivation, we derive the monomial equivalent codes for these repeated-root cyclic codes with prime power lengths. Given that monomial equivalent codes exhibit identical weight distributions, we transform the computation of the weight distribution of these repeated-root cyclic codes into the computation of the weight distribution of their monomial equivalent codes. Leveraging the classical results on the weight distribution of MDS codes, we explicitly determine the weight distribution of these repeated-root cyclic codes. Moreover, we apply the weight distribution formula to construct a class of $p$-weight cyclic codes for any prime $p$.

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arXiv2025

Self-orthogonal codes from plateaued functions and their applications in quantum codes and LCD codes

Self-orthogonal codes have received great attention due to their important applications in quantum codes, LCD codes and lattices. Recently, several families of self-orthogonal codes containing the all-$1$ vector were constructed by augmentation technique. In this paper, utilizing plateaued functions, we construct some classes of linear codes which do not contain the all-$1$ vector. We also investigate their punctured codes. The weight distributions of the constructed codes are explicitly determined. Under certain conditions, these codes are proved to be self-orthogonal. Furthermore, some classes of optimal linear codes are obtained from their duals. Using the self-orthogonal punctured codes, we also construct several new families of at least almost optimal quantum codes and optimal LCD codes.

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arXiv2025

Multilevel inserting constructions for constant dimension subspace codes

Subspace codes, especially constant dimension subspace codes (CDCs), represent an intriguing domain that can be used to conduct basic coding theory investigations. They have received widespread attention due to their applications in random network coding. This paper presents inverse bilateral multilevel construction by introducing inverse bilateral identifying vectors and inverse bilateral Ferrers diagram rank-metric codes. By inserting the inverse bilateral multilevel construction into the double multilevel construction and bilateral multilevel construction, an effective construction for CDCs is provided. Furthermore, via providing a new set of bilateral identifying vectors, we give another efficient construction for CDCs. In this article, several CDCs are exhibited, equipped with the rank-metric, with larger sizes than the known ones in the existing literature. From a practical standpoint, our results could help in the pragmatic framework of constant-dimension-lifted rank-metric codes for applications in network coding. The ratio of the new lower bound to the known upper bound for some CDCs is calculated, which is greater than 0.94548 for any prime power $q \geq 3.$

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arXiv2025

New constant-dimension subspace codes from parallel cosets of optimal Ferrers diagram rank-metric codes and multilevel inserting constructions

Constant-dimension subspace codes (CDCs), a special class of subspace codes, have attracted significant attention due to their applications in network coding. A fundamental research problem of CDCs is to determine the maximum number of codewords under the given parameters. The paper first proposes the construction of parallel cosets of optimal Ferrers diagram rank-metric codes (FDRMCs) by employing the list of CDCs and inverse list of CDCs. Then a new class of CDCs is obtained by combining the parallel cosets of optimal FDRMCs with parallel linkage construction. Next, we present a novel set of identifying vectors and provide a new construction of CDCs via the multilevel constuction. Finally, the coset construction is inserted into the multilevel construction and three classes of large CDCs are provided, one of which is constructed by using new optimal FDRMCs. Our results establish at least 65 new lower bounds for CDCs with larger sizes than the previously best known codes.

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arXiv2025

Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction

Constant dimension codes (CDCs), as special subspace codes, have received extensive attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximal possible size $A_q(n,d,\{k\})$ for given parameters $q, n, d$, and $k$. This paper introduces criteria for choosing appropriate bilateral identifying vectors compatible with the parallel mixed dimension construction (Des. Codes Cryptogr. 93(1):227--241, 2025). We then utilize the generalized bilateral multilevel construction (Des. Codes Cryptogr. 93(1):197--225, 2025) to improve the parallel mixed dimension construction efficiently. Many new CDCs that are better than the previously best-known codes are constructed.

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arXiv2025

Decentralized Multi-Authority Attribute-Based Inner-Product Functional Encryption: Noisy and Evasive Constructions from Lattices

We study multi-authority attribute-based functional encryption for noisy inner-product functionality, and propose two new primitives: (1) multi-authority attribute-based (noisy) inner-product functional encryption (MA-AB(N)IPFE), which generalizes existing multi-authority attribute-based IPFE schemes by Agrawal et al. (TCC'21), by enabling approximate inner-product computation; and (2) multi-authority attribute-based evasive inner-product functional encryption (MA-evIPFE), a relaxed variant inspired by the evasive IPFE framework by Hsieh et al. (EUROCRYPT'24), shifting focus from ciphertext indistinguishability to a more relaxed pseudorandomness-based security notion. To support the above notions, we introduce two variants of lattice-based computational assumptions: evasive IPFE assumption and indistinguishability-based evasive IPFE assumption (IND-evIPFE). We present lattice-based constructions of both primitives for subset policies, building upon the framework of Waters et al.( TCC'22). Our schemes are proven to be statically secure in the random oracle model under the standard LWE assumption and the newly introduced assumptions. Additionally, we show our MA-AB(N)IPFE scheme can be transformed via modulus switching into a noiseless MA-IPFE scheme that supports exact inner-product functionality. This yields the first lattice-based construction of such a primitive. All our schemes support arbitrary polynomial-size attribute policies and are secure in the random oracle model under lattice assumptions with a sub-exponential modulus-to-noise ratio, making them practical candidates for noise-tolerant, fine-grained access control in multi-authority settings.

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arXiv2025

Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

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arXiv2025

New Constant Dimension Codes From the Inserting Mixed Dimension Construction and Multilevel Construction

Constant dimension codes (CDCs) are essential for error correction in random network coding. A fundamental problem of CDCs is to determine their maximal possible size for given parameters. Inserting construction and multilevel construction are two effective techniques for constructing CDCs. We first provide a sufficient condition for a subspace to be added to the code from the mixed dimension construction in Lao et al. (IEEE Trans. Inf. Theory 69(7): 4333-4344, 2023). By appropriately combining matrix blocks from small CDCs and rank-metric codes, we introduce three inserting constructions based on the mixed dimension construction. Furthermore, the mixed dimension construction and these inserting constructions are improved by the multilevel construction that is based on lifting rank-restricted Ferrers diagram rank-metric codes. Our constructions yield some new lower bounds for CDCs, which are superior to the previously best-known ones.

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arXiv2024

Association schemes arising from non-weakly regular bent functions

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 [13], {Ö}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-weakly regular bent functions.They also stated that constructing $2p$-class association schemes from $p$-ary non-weakly regular bent functions is an interesting problem, where $p>3$ is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$ for any odd prime $p$. Fusing those association schemes, we also obtain $t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we give the sufficient and necessary conditions for the partitions $P$, $D$, $T$, $U$ and $V$ (defined in this paper) to induce symmetric association schemes.

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arXiv2024

Decoding Error Probability of the Random Matrix Ensemble over the Erasure Channel

Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random matrix ensemble and the error exponent of the variance, which imply a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

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arXiv2024

Self-Orthogonal Codes from Vectorial Dual-Bent Functions

Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In \cite{HLL2023Te,LH2023Se}, $p$-ary self-orthogonal codes were constructed by using $p$-ary weakly regular bent functions, where $p$ is an odd prime. In \cite{WH2023Se}, two classes of non-degenerate quadratic forms were used to construct $q$-ary self-orthogonal codes, where $q$ is a power of a prime. In this paper, we construct new families of $q$-ary self-orthogonal codes using vectorial dual-bent functions. Some classes of at least almost optimal linear codes are obtained from the dual codes of the constructed self-orthogonal codes. In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes. From the view of vectorial dual-bent functions, we illustrate that the works on constructing self-orthogonal codes from $p$-ary weakly regular bent functions \cite{HLL2023Te,LH2023Se} and non-degenerate quadratic forms with $q$ being odd \cite{WH2023Se} can be obtained by our results. We partially answer an open problem on determining the weight distribution of a class of self-orthogonal codes given in \cite{LH2023Se}. As applications, we construct new infinite families of at least almost optimal $q$-ary linear complementary dual codes (for short, LCD codes) and quantum codes.

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arXiv2023

Linear codes with few weights from non-weakly regular plateaued functions

Linear codes with few weights have significant applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. There are a number of methods to construct linear codes, one of which is based on functions. Furthermore, two generic constructions of linear codes from functions called the first and the second generic constructions, have aroused the research interest of scholars. Recently, in \cite{Nian}, Li and Mesnager proposed two open problems: Based on the first and the second generic constructions, respectively, construct linear codes from non-weakly regular plateaued functions and determine their weight distributions. Motivated by these two open problems, in this paper, firstly, based on the first generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions and determine the weight distributions of the constructed codes. Next, based on the second generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions belonging to $\mathcal{NWRF}$ (defined in this paper) and determine the weight distributions of the constructed codes. We also give the punctured codes of these codes obtained based on the second generic construction and determine their weight distributions. Meanwhile, we obtain some optimal and almost optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that the constructed codes are minimal for almost all cases and obtain some secret sharing schemes with nice access structures based on their dual codes.

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arXiv2023

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently, locally repairable codes (LRCs) have attracted much attention due to their repair efficiency in large-scale distributed storage systems. In this paper, by employing the singleton procedure technique, we first provide a sufficient condition for bounding the minimum distance of cyclic codes with typical defining sets. Secondly, by considering a specific case, we establish a connection between bounds for the minimum distance of cyclic codes and solutions to a system of inequalities. This connection leads to the derivation of new bounds, including some with general patterns. In particular, we provide three new bounds with general patterns, one of which serves as a generalization of the Betti-Sala bound. Finally, we present a generalized lower bound for a special case and construct several families of $(2, δ)$-LRCs with unbounded length and minimum distance $2δ$. It turns out that these LRCs are distance-optimal, and their parameters are new. To the best of our knowledge, this work represents the first construction of distance-optimal $(r, δ)$-LRCs with unbounded length and minimum distance exceeding $r+δ-1$.

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