arXiv subjects

Chang-An Zhao

Publications and source records attributed to Chang-An Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Self-Dual Cyclic Codes with Improved Minimum Distance Estimates via Extending the Chen-Ding Construction

Self-dual cyclic codes have garnered significant interest owing to their rich algebraic structures and wide-ranging applicability. Their construction and the establishment of lower bounds on their minimum distances are fundamental problems in coding theory. Chen and Ding laid an important foundation for the construction of self-dual cyclic codes in the case where the multiplicative order of $q$ module $n$, denoted by $\operatorname{ord}_n(q)$, is odd. Building on their work, we extend the investigation to the case of even order $\operatorname{ord}_n(q)$ and demonstrate that the minimum distances of the resulting self-dual cyclic codes satisfy square-root lower bounds. By examining the consecutive zero segments in the defining set of the dual code, we determine the exact parameters of Euclidean self-dual cyclic codes with even $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with odd $\operatorname{ord}_n(q)$. Furthermore, for Euclidean self-dual cyclic codes with odd $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with even $\operatorname{ord}_n(q)$, we introduce a refined parameter selection that leads to larger minimum distances with the same code length and dimension. This approach also yields tighter lower bounds for several families of self-dual cyclic codes. This work enriches the theory of self-dual cyclic codes and offers new insights into estimating lower bounds on their minimum distances.

cs.IT↗

arXiv2026

Rank-Two Drinfeld Module over Elliptic Curves

Drinfeld modules, introduced by D.~V.~Drinfeld in the 1970s, were originally developed as a function field analogue of elliptic curves and have since become a central tool in the Langlands program over function fields. The theory has been highly developed and shares deep connections with many areas, including algebraic geometry, number theory, and coding theory. Despite these advances, the explicit construction of Drinfeld modules over non-polynomial coordinate rings remains a largely open problem. Indeed, aside from the classical polynomial case $\mathbb{F}_q[t]$, explicit formulas for Drinfeld modules are known only in very limited situations. Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ with a fixed rational point $\infty$, and let $\mathbf{A} = H^0(E\setminus\{\infty\}, \mathcal{O}_E)$ be its coordinate ring. Rank-one Drinfeld $ \mathbf{A} $-modules were explicitly constructed by Green and Papanikolas, providing the first systematic example beyond the polynomial case. However, the rank-two case has remained completely inaccessible until now. This paper solves the rank-two case of this open problem in a fully explicit manner. Precisely, we develop an explicit theory for rank-two sign-normalized $\mathbf{A}$-Drinfeld modules over an algebraically closed $\mathbf{A}$-field $L$, building upon the rank-one framework. We determine the structure of the associated Anderson motive $M_ϕ$ and prove that it is generated by three elements subject to a single quadratic $τ$-relation, which we derive in closed form using the geometric parameters of the underlying elliptic curve.As a consequence, we find that the moduli space of sign-normalized rank-two Drinfeld $\mathbf{A}$-modules is an open domain $Y\neq 0 $ inside a supersingular curve $Y^{q+1} = π(X) $, where $π$ is an explicit polynomial of degree $2q+1$.

math.NT↗

arXiv2026

Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions

Recently, constructions of linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes on function fields have attracted considerable attention due to the wide range of applications of these codes. Such constructions rely on non-special divisors of degrees $g$ and $g-1$. In this work, we investigate Kummer extensions defined by $y^m = f(x)$ with $f(x)\in\mathbb{F}_q(x)$ and establish an arithmetic characterization of non-special divisors whose support can contain non-totally ramified places. Based on this characterization, we explicitly construct non-special divisors of degree $g-1$ on the GK curve. Moreover, utilizing pure gaps, we explicitly provide several families of effective non-special divisors of degree $g$ on Kummer extensions with the same multiplicities. We then develop a general framework for constructing LCPs of algebraic geometry (AG) codes on Kummer extensions. By virtue of canonical divisors, we show that the security parameters of LCPs of AG codes can be determined within this framework, which also enables the construction of LCD AG codes. Finally, we illustrate our results with representative examples, including LCPs of codes on the GK curve and LCD codes on quotients of the Hermitian curve.

cs.IT↗

arXiv2026

On the Maximal Length of MDS Elliptic Codes

The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the maximal length of a $q$-ary MDS elliptic code of dimension $k$. It was recently shown that $\operatorname{MEC}(k,q)\le\frac{q+1}{2}+\sqrt{q}$ for $q\ge289$ and $3\le k\le(q+1-2\sqrt{q})/10$, with equality for odd $k$ when $q$ is an odd square. This paper investigates the remaining open cases, namely even dimension $k$, non-square $q$ and fields of characteristic $2$, and provides a complete resolution of the tightness question for the two natural parity regimes of $q+1+\lfloor 2\sqrt{q}\rfloor$. We prove that if the support of $G$ (used to define the code) consists of $\mathbb{F}_q$-rational points, the bound decreases to $\frac{q+1}{2}+\sqrt{q}-1$ for even $k$. Without this restriction, we construct MDS codes attaining $\frac{q+1}{2}+\sqrt{q}$ for even $k$. More generally, we establish $\operatorname{MEC}(k,q)=\frac{q+1+\lfloor2\sqrt{q}\rfloor}{2}$ when $q+1+\lfloor2\sqrt{q}\rfloor$ is even, and $\operatorname{MEC}(k,q)=\frac{q+\lfloor2\sqrt{q}\rfloor}{2}$ when it is odd.

cs.IT↗

arXiv2026

Weierstrass semigroups at totally ramified places of degree one on linearized function fields

A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$. As a consequence, we explicitly provide a system of generators, the multiplicity, and the Frobenius number of the Weierstrass semigroup $H(Q)$. Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric. Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we investigate the linearized function field associated with the denominator of a separable polynomial and apply our results to present several examples.

math.NT↗

arXiv2026

On Permutation Groups of Cyclic Codes over Finite Fields

The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic codes with very long lengths and special generator polynomials to those with prime lengths. Consequently, we mainly determine the permutation groups of certain cyclic codes over $\mathbb{F}_{r^α}$ with lengths $hp$, $r^mp^n$ and $pq$ and special generator polynomials where $h$ is a positive integer and $p$, $q$ and $r$ are distinct prime numbers. For length $pq$, we manage to provide the permutation groups of cyclic codes with generator polynomials $Q_{pq}(x)$(the $pq$-th cyclotomic polynomial) or others, which seems to be the first work about permutation groups of cyclic codes with generator polynomials that are factors of $x^{pq}-1$ but not factors of $x^p-1(\text{or }x^q-1)$.

cs.IT↗

arXiv2026

Weierstrass semigroups at totally ramified places of degree one on Kummer extensions

We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation $y^m = f(x)$ over $K$, an algebraic extension of $\mathbb{F}_q$, where $f(x)\in K(x)$. Our description takes a unified form for distinct totally ramified places of degree one. We then provide a necessary and sufficient condition for the Weierstrass semigroup at a totally ramified place of degree one to be symmetric. Furthermore, we investigate the minimal generating set of the Weierstrass semigroups at many totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also construct functions whose pole divisors have coefficients lying in the set. Finally, we apply our results to specific Kummer extensions, including function fields of GGS curves and subcovers of the BM curve.

math.AG↗

arXiv2026

Locally Repairable Codes with Availability via Elliptic Function Fields

Locally repairable codes with availability have become essential components in modern large-scale distributed cloud storage systems and numerous other applications. In this paper, we focus on the construction of locally repairable codes with one or two recovering sets via elliptic function fields. Prior pioneering work by Li et al. (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) employed maximal supersingular elliptic curves to obtain several optimal (classical) locally repairable codes. In contrast, we consider ordinary elliptic curves with many rational points. This approach yields several new families of $q$-ary optimal locally repairable codes with length $O(q+2\sqrt{q})$ and flexible locality. Consequently, our work broadens the selection of curves available for the construction of optimal locally repairable codes. Furthermore, we present a general framework for constructing locally repairable codes with two recovering sets via automorphism groups of elliptic function fields. To realize this framework, we devise a novel construction for determining the functions $e_i$ in the construction of locally repairable codes. By employing both supersingular and ordinary elliptic curves, we obtain several families of locally repairable codes with two recovering sets. In particular, we construct a family of $q^2$-ary locally repairable codes with two recovering sets, achieving length $O(q^2+2q)$ and Singleton-defect $O\!\left(\frac{2\ell}{q^2+2q-8\ell}\right)$, where $\ell \mid\mid q + 2$ with $4\ell < q$.

cs.IT↗

arXiv2026

Successive Cancellation List Decoding of Extended Reed-Solomon Codes

Reed-Solomon (RS) codes are an important class of non-binary error-correction codes. They are particularly competent in correcting burst errors, being widely applied in modern communications and data storage systems. This also thanks to their distance property of reaching the Singleton bound, being the maximum distance separable (MDS) codes. This paper proposes a new list decoding for extended RS (eRS) codes defined over a finite field of characteristic two, i.e., F_{2^n}. It is developed based on transforming an eRS code into n binary polar codes. Consequently, it can be decoded by the successive cancellation (SC) decoding and further their list decoding, i.e., the SCL decoding. A pre-transformed matrix is required for reinterpretating the eRS codes, which also determines their SC and SCL decoding performances. Its column linear independence property is studied, leading to theoretical characterization of their SC decoding performance. Our proposed decoding and analysis are validated numerically.

cs.IT↗

arXiv2025

Distributed Matrix Multiplication-Friendly Algebraic Function Fields

In this paper, we introduce distributed matrix multiplication (DMM)-friendly algebraic function fields for polynomial codes and Matdot codes, and present several constructions for such function fields through extensions of the rational function field. The primary challenge in extending polynomial codes and Matdot codes to algebraic function fields lies in constructing optimal decoding schemes. We establish optimal recovery thresholds for both polynomial algebraic geometry (AG) codes and Matdot AG codes for fixed matrix multiplication. Our proposed function fields support DMM with optimal recovery thresholds, while offering rational places that exceed the base finite field size in specific parameter regimes. Although these fields may not achieve optimal computational efficiency, our results provide practical improvements for matrix multiplication implementations. Explicit examples of applicable function fields are provided.

cs.IT↗

arXiv2025

Dual and Covering Radii of Extended Algebraic Geometry Codes

Many literatures consider the extended Reed-Solomon (RS) codes, including their dual codes and covering radii, but few focus on extended algebraic geometry (AG) codes of genus $g\ge1$. In this paper, we investigate extended AG codes and Roth-Lempel type AG codes, including their dual codes and minimum distances. Moreover, we show that for certain $g$, the length of a $g$-MDS code over a finite field $\mathbb{F}_q$ can attain $q+1+2g\sqrt{q}$, which is achieved by an extended AG code from the maximal curves of genus $g$. Notably, for some small finite fields, this length $q+1+2g\sqrt{q}$ is the largest among all known $g$-MDS codes. Subsequently, we establish that the covering radius of an $[n,k]$ extended AG code has $g+2$ possible values. For the case of $g=1$, we prove that this range reduces to two possible values when the length $n$ is sufficiently large, or when there exists an $[n,k+1]$ MDS elliptic code.

cs.IT↗

arXiv2025

Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions

For a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{λ_i}$ over an algebraic extension $K$ of a finite field $\fq$, where $\la_i\in \Z\backslash\{0\}$ for $1\leq i\leq r$, $\gcd(m,q) = 1$, and $\a_1,\cdots,\a_r\in K$ are pairwise distinct elements, we propose a simple and efficient method to find all pure gaps at many totally ramified places. We introduce a bottom set of pure gaps and indicate that the set of pure gaps is completely determined by the bottom set. Furthermore, we demonstrate that a pure gap can be deduced from a known pure gap by easily verifying only one inequality. Then, in the case where $λ_1 = λ_2 = \cdots = λ_r$, we fully determine an explicit description of the set of pure gaps at many totally ramified places, This includes the scenario in which the set of these places contains the infinite place. Finally, we apply these results to construct multi-point algebraic geometry codes with good parameters. As one of the examples, a presented code with parameters $[74, 60, \geq 10]$ over $\mathbb{F}_{25}$ yields a new record.

cs.IT↗

arXiv2025

Optimal and Almost Optimal Locally Repairable Codes from Hyperelliptic Curves

Locally repairable codes are widely applicable in contemporary large-scale distributed cloud storage systems and various other areas. By making use of some algebraic structures of elliptic curves, Li et al. developed a series of $q$-ary optimal locally repairable codes with lengths that can extend to $q+2\sqrt{q}$. In this paper, we generalize their methods to hyperelliptic curves of genus $2$, resulting in the construction of several new families of $q$-ary optimal or almost optimal locally repairable codes. Our codes feature lengths that can approach $q+4\sqrt{q}$, and the locality can reach up to $239$.

cs.IT↗

arXiv2025

On Iso-Dual MDS Codes From Elliptic Curves

For a linear code $C$ over a finite field, if its dual code $C^{\perp}$ is equivalent to itself, then the code $C$ is said to be {\it isometry-dual}. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length $n$ satisfies $n\le\frac{q+\lfloor2\sqrt{q}\rfloor-1}{2}$ when $q$ is even and $n\le\frac{q+\lfloor2\sqrt{q}\rfloor-3}{2}$ when $q$ is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.

cs.IT↗

arXiv2025

Extension of Optimal Locally Repairable codes

Recent studies have delved into the construction of locally repairable codes (LRCs) with optimal minimum distance from function fields. In this paper, we present several novel constructions by extending the findings of optimally designed locally repairable codes documented in the literature. Let $C$ denote an optimal LRC of locality $r$, implying that every repairable block of $C$ is a $[r+1, r]$ MDS code, and $C$ maximizes its minimum distance. By extending a single coordinate of one of these blocks, we demonstrate that the resulting code remains an optimally designed locally repairable code. This suggests that the maximal length of an optimal LRC from rational function fields can be extended up to $q+2$ over a finite field $\mathbb{F}_q$. In addition, we give a new construction of optimal $(r, 3)$-LRC by extending one coordinate in each block within $C$. Furthermore, we propose a novel family of LRCs with Roth-Lempel type that are optimal under certain conditions. Finally, we explore optimal LRCs derived from elliptic function fields and extend a single coordinate of such codes. This approach leads us to confirm that the new codes are also optimal, thereby allowing their lengths to reach $q + 2\sqrt{q} - 2r - 2$ with locality $r$. We also consider the construction of optimal $(r, 3)$-LRC in elliptic function fields, with exploring one more condition.

cs.IT↗

arXiv2022

Linear Complexity of A Family of Binary $pq^2$-periodic Sequences From Euler Quotients

We first introduce a family of binary $pq^2$-periodic sequences based on the Euler quotients modulo $pq$, where $p$ and $q$ are two distinct odd primes and $p$ divides $q-1$. The minimal polynomials and linear complexities are determined for the proposed sequences provided that $2^{q-1} \not\equiv 1 \mod{q^2}.$ The results show that the proposed sequences have high linear complexities.

cs.IT↗

arXiv2021

An Alternative Approach for Computing Discrete Logarithms in Compressed SIDH

Currently, public-key compression of supersingular isogeny Diffie-Hellman (SIDH) and its variant, supersingular isogeny key encapsulation (SIKE) involve pairing computation and discrete logarithm computation. In this paper, we propose novel methods to compute only 3 discrete logarithms instead of 4, in exchange for computing a lookup table efficiently. The algorithms also allow us to make a trade-off between memory and efficiency. Our implementation shows that the efficiency of our algorithms is close to that of the previous work, and our algorithms perform better in some special cases.

cs.CR↗

arXiv2021

The Elliptic Net Algorithm Revisited

Pairings have been widely used since their introduction to cryptography. They can be applied to identity-based encryption, tripartite Diffie-Hellman key agreement, blockchain and other cryptographic schemes. The Acceleration of pairing computations is crucial for these cryptographic schemes or protocols. In this paper, we will focus on the Elliptic Net algorithm which can compute pairings in polynomial time, but it requires more storage than Miller's algorithm. We use several methods to speed up the Elliptic Net algorithm. Firstly, we eliminate the inverse operation in the improved Elliptic Net algorithm. In some circumstance, this finding can achieve further improvements. Secondly, we apply lazy reduction technique to the Elliptic Net algorithm, which helps us achieve a faster implementation. Finally, we propose a new derivation of the formulas for the computation of the Optimal Ate pairing on the twisted curve. Results show that the Elliptic Net algorithm can be significantly accelerated especially on the twisted curve. The algorithm can be $80\%$ faster than the previous ones on the twisted 381-bit BLS12 curve and $71.5\%$ faster on the twisted 676-bit KSS18 curve respectively.

cs.CR↗