Algebraic Geometry Codes Research Articles

Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field Fq2 is maximal if the number Nq2 of its Fq2-rational points attains the Hasse-Weil upper bound, that is Nq2=q2+2gq+1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p2 where p is the characteristic of Fq2. Doing so we also determine the Fq2-isomorphism classes of such curves and describe their full Fq2-automorphism groups.

Bases for Riemann–Roch spaces of linearized function fields with applications to generalized algebraic geometry codes

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters.

Algebraic Geometry Codes in the Sum–Rank Metric

Algebraic Geometry Codes in the Sum–Rank Metric

On linear codes with random multiplier vectors and the maximum trace dimension property

Abstract Let C C be a linear code of length n n and dimension k k over the finite field F q m {{\mathbb{F}}}_{{q}^{m}} . The trace code Tr ( C ) {\rm{Tr}}\left(C) is a linear code of the same length n n over the subfield F q {{\mathbb{F}}}_{q} . The obvious upper bound for the dimension of the trace code over F q {{\mathbb{F}}}_{q} is m k mk . If equality holds, then we say that C C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a {C}_{{\boldsymbol{a}}} denote the code obtained from C C and a multiplier vector a ∈ ( F q m ) n {\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n} . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a {C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m ( k + h ) n\ge m\left(k+h) , where h ≥ 0 h\ge 0 is the Singleton defect of C C . For the extremal case n = m ( h + k ) n=m\left(h+k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

New dimension-independent upper bounds on linear insdel codes

To determine the insdel distances of linear codes is a very challenging problem. The half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li is a basic upper bound on the insertion-deletion error-correcting capabilities of linear codes. In this paper we give several new coordinate ordering-free and coordinate ordering-depending upper bounds for the insdel distances of linear codes. These upper bounds do not depend on dimensions of linear codes and only depend on the formations of codewords. It is shown that for many natural well-known linear codes including binary simplex codes, Kasami codes, many linear binary codes with few non-zero weights and some algebraic geometry codes, the new coordinate ordering-free upper bounds on their insdel distances are strictly smaller than the half-Singleton bound and the direct upper bound. On the other hand for many linear binary codes, the ordering-depending upper bound on their insdel distance is 2. We also give ordering-depending upper bounds on the insdel distances of some linear ternary codes with few non-zero weights and some binary and ternary algebraic geometry codes.

A New Construction of Nonlinear Codes via Algebraic Function Fields

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. A great many good codes have been constructed over alphabets of sizes equal to prime powers, however, good block codes over other alphabet sizes are rare. In this paper, we provide a new explicit construction of ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> + 1)-ary nonlinear codes via algebraic function fields, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is a prime power. Our codes are constructed by evaluating rational functions at all rational places of an algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i></sub> ⋃ {∞}. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.

Алгеброгеометрические коды и декодирование на основе пар, исправляющих ошибки

We consider the basic theory of algebraic curves and their function fields necessary for constructing algebraic geometry codes and a pair of codes forming an error-correction pair which is used in a precomputation step of the decoding algorithm for the algebraic geometry codes. Also, we consider the decoding algorithm and give the necessary theory to prove its correctness. As a result, we consider elliptic curves, Hermitian curves and Klein quartics and construct the algebraic geometry codes associated with these families of curves, and also explicitly define the error-correcting pairs for the resulting codes.

A representation of Galois dual codes of algebraic geometry codes via Weil differentials

Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the &lt;inline-formula&gt;&lt;tex-math id="M910"&gt;\begin{document}$ h $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;-Galois dual code of an algebraic geometry code &lt;inline-formula&gt;&lt;tex-math id="M900"&gt;\begin{document}$ C_{ {\cal{L}},F}(D,G) $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; from function field &lt;inline-formula&gt;&lt;tex-math id="M904"&gt;\begin{document}$ F/ \mathbb{F}_{p^e} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; can be represented as an algebraic geometry code &lt;inline-formula&gt;&lt;tex-math id="M902"&gt;\begin{document}$ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; from an associated function field &lt;inline-formula&gt;&lt;tex-math id="M903"&gt;\begin{document}$ F'/ \mathbb{F}_{p^e} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with an isomorphism &lt;inline-formula&gt;&lt;tex-math id="M600"&gt;\begin{document}$\phi_{h}:F\rightarrow F'$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; satisfying &lt;inline-formula&gt;&lt;tex-math id="M700"&gt;\begin{document}$ \phi_{h}(a) = a^{p^{e-h}} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for all &lt;inline-formula&gt;&lt;tex-math id="M800"&gt;\begin{document}$ a\in \mathbb{F}_{p^e} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. As an application of this result, we construct a family of Galois linear complementary dual maximum distance separable codes (LCD MDS codes).

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.

On the automorphisms of generalized algebraic geometry codes

On the automorphisms of generalized algebraic geometry codes

MDS linear codes with one-dimensional hull

The hull of a linear code C is the intersection of C with its dual C⊥, where the dual is often defined with respect to Euclidean or Hermitian inner product. The Euclidean hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. Recently, both Euclidean and Hermitian hulls have found another application to quantum error correcting codes with entanglements. This paper aims to explore explicit constructions of families of MDS linear codes with one-dimensional hull for both cases. We use tools from algebraic function fields in one variable to study such codes. Sufficient conditions for an algebraic geometry code of genus zero to have one-dimensional hull are provided, and some construction methods are presented. We construct many families of MDS linear codes with one-dimensional hull for the Euclidean case and three families for the Hermitian case, respectively.

Bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes

We construct explicit bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes with good parameters. This correspondence is a generalization of a work of Maharaj, Matthews, and Pirsic.

Two-point AG codes from the Beelen-Montanucci maximal curve

In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes. AG codes with better parameters with respect to comparable two-point codes from the Garcia-Güneri-Stichtenoth (GGS) curve are discovered.

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field ${\mathcal F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of ${\mathcal F}$ and $P, Q$ are not in $\mbox{supp}(D)$. We characterize those sequences in terms of $b$ for general function fields. We then apply the result to the broad class of Kummer extensions ${\mathcal F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach

Support constrained generator matrix for a linear code has been an active topic in recent years. The necessary and sufficient condition for the existence of MDS codes over small fields with support constrained generator matrices were conjectured by Dau, Song, Yuen in 2014. This GM-MDS conjecture was proved independently by Lovett and Yildiz-Hassibi in 2018. In this paper we propose the necessary and sufficient conditions for support constrained generator matrices of general linear codes based on the generalized Hamming weights. It is proved that the direct generalization of the GM-MDS conjecture for $2$-MDS codes and algebraic geometry codes is not true over arbitrary fields. We propose and prove a GHW-based sufficient condition for support constrained matrices of general linear codes. This is the first sufficient condition for the existence of support constrained generator matrices for general linear codes over arbitrary finite fields. Moreover a weaker GHW-based sufficient condition for support constrained generator matrices are given for the binary simplex code and the first order $q$-ary Reed-Muller codes.

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes

In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdon–Bezerra–Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernandez.

A comparison between Suzuki invariant code and one-point Hermitian code

In this paper we compare the performance of two algebraic geometry codes (Suzuki and Hermitian codes) constructed using maximal algebraic curves over [Formula: see text] with large automorphism groups by choosing specific divisors. We discuss their parameters, compare the rate of these codes as well as their relative minimum distances, and we show that both codes are asymptotically good in terms of the rate which is in contrast to their behavior in terms of the relative minimum distance.

New families of self-dual codes

Recently, the author has constructed families of MDS Euclidean self-dual codes from genus zero algebraic geometry (AG) codes. In the present correspondence, more families of optimal Euclidean self-dual codes from AG codes are explored. New families of MDS Euclidean self-dual codes of odd characteristic and those of almost MDS Euclidean self-dual codes are constructed explicitly from genus zero and genus one curves, respectively. More families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.

Several constructions of LCD MDS codes

Linear complementary dual codes (LCD codes) have been widely studied because of their significant role in data storage and cryptography. Maximum distance separable codes with complimentary duals (LCD MDS codes) have the greatest error correcting ability, and so the construction of LCD MDS codes is a hot topic. By using the generator matrix of linear codes, the existence results of LCD MDS codes are given by Carlet et al. in 2018. Recently, several display constructions of LCD MDS codes are given by scholars via the generalized Reed-Solomoncodes, the algebraic geometry codes, the constacylic codes and so on. In this paper, we will introduce the main methods, results and recent progress of constructing LCD MDS codes.

О ГРАНИЦАХ МОЩНОСТИ ЗЛОУМЫШЛЕННИКОВ ДЛЯ ИДЕНТИФИЦИРУЮЩИХ АЛГЕБРОГЕОМЕТРИЧЕСКИХ КОДОВ НА СПЕЦИАЛЬНЫХ КРИВЫХ

Broadcast encryption is a data distribution protocol which can prevent malefactor parties from unauthorized accessing or copying the distributed data. It is widely used in distributed storage and network data protection schemes. To block the socalled coalition attacks on the protocol, classes of error-correcting codes with special properties are used, namely c-FP and c-TA properties. We study the problem of evaluating the lower and the upper boundaries on coalition power, within which the algebraic geometry codes possess these properties. Earlier, these boundaries were calculated for single-point algebraic-geometric codes on curves of the general form. Now, we clarified these boundaries for single-point codes on curves of a special form; in particular, for codes on curves on which there are many equivalence classes after factorization by equality of the corresponding points coordinates relation.