Recent zbMATH articles in MSC 14G50https://www.zbmath.org/atom/cc/14G502021-04-16T16:22:00+00:00WerkzeugLCD codes and self-orthogonal codes in generalized dihedral group algebras.https://www.zbmath.org/1456.140332021-04-16T16:22:00+00:00"Gao, Yanyan"https://www.zbmath.org/authors/?q=ai:gao.yanyan"Yue, Qin"https://www.zbmath.org/authors/?q=ai:yue.qin"Wu, Yansheng"https://www.zbmath.org/authors/?q=ai:wu.yanshengA group code is a right ideal in a group ring \(R[G]\) where \(R\) is a commutative ring and \(G\) a finite group. In this paper, the authors consider a finite field \(\mathbb{F}_q\) and a generalized dihedral group \(D_{2n,r}\), \(\gcd(2n,q)=1\). As a first main result, they explicitly describe the primitive idempotents of the group algebra \(\mathbb{F}_q[D_{2n,r}]\). Given a code \(C\), it is named LCD whenever \(C \cap C^\perp = \{0\}\) and it is self-orthogonal iff \(C \subseteq C^\perp\), where \(C^\perp\) means dual of \(C\). LCD codes have cryptographic applications and self-orthogonal codes provide quantum codes. The second main result of the paper describes and counts LCD and self-orthogonal group codes in \(\mathbb{F}_q[D_{2n,r}]\).
Reviewer: Carlos Galindo (Castellón)Counting Richelot isogenies between superspecial abelian surfaces.https://www.zbmath.org/1456.140552021-04-16T16:22:00+00:00"Katsura, Toshiyuki"https://www.zbmath.org/authors/?q=ai:katsura.toshiyuki"Takashima, Katsuyuki"https://www.zbmath.org/authors/?q=ai:takashima.katsuyukiSummary: \textit{W. Castryck} et al. [``Hash functions from superspecial genus-2 curves using Richelot isogenies'', J. Math. Crypt. 14, 268--292 (2020; \url{doi:10.1515/jmc-2019-0021})] used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, \textit{C. Costello} and \textit{B. Smith} [``The supersingular isogeny problem in genus 2 and beyond'', in: Post-quantum cryptography. Cham: Springer. 151--168 (2020)], devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of decomposed Richelot isogenies in terms of involutive reduced automorphisms of genus-2 curves over a finite field, and explicitly count such decomposed (and nondecomposed) Richelot isogenies between superspecial principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al. [loc. cit.].
For the entire collection see [Zbl 1452.11005].Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus.https://www.zbmath.org/1456.140562021-04-16T16:22:00+00:00"Aubry, Yves"https://www.zbmath.org/authors/?q=ai:aubry.yves"Berardini, Elena"https://www.zbmath.org/authors/?q=ai:berardini.elena"Herbaut, Fabien"https://www.zbmath.org/authors/?q=ai:herbaut.fabien"Perret, Marc"https://www.zbmath.org/authors/?q=ai:perret.marcIn this paper, a theoretical study of algebrao-geometric codes constructed from abelian surfaces \(A\) defined over finite fields \(F_q\) is provided. Let \(m = \lfloor 2 \sqrt{q}\rfloor\), \(H\) be an ample divisor on \(A\) rational over \(F_q\) and \(r\) be a positive integer such that \(rH\) is very ample. We denote by \(Tr(A)\) the trace of \(A\) and by \(C(A, rH)\) the generalised evaluation code. Then, it is proved that the distance \(d(A, rH)\) of the code \(C(A, rH)\) satisfies the inequality: \[d(A, rH) \geq \# A(F_q) - rH^2(q + 1 -Tr(A) + m) - r^2m \frac{H^2}{2}.\] Furthermore, if \(A\) is simple and contains no absolutely irreducible curves of arithmetic genus \(\leq \ell\), then \[d(A, rH) \geq \# A(F_q)-\max\left( r\left\lfloor \sqrt{\frac{H^2}{2}}\right \rfloor (\ell-1), \varphi(1), \varphi\left(\left\lfloor r \sqrt{\frac{H^2}{2 \ell}}\right\rfloor \right) \right), \] where \[ \varphi(x) := m \left( r \sqrt{\frac{H^2}{2}} - x\sqrt{\ell} \right)^2 +2m\ell \left( r \sqrt{\frac{H^2}{2}} - x\sqrt{\ell} \right)+\] \[ x\Big(q+1-Tr(A)+(\ell-1)(m-\sqrt{\ell})\Big)+r\sqrt{\frac{H^2}{2}}\ (\ell-1).\] The proof of this result is relied on a characterisation of isogeny classes of abelian surfaces containing Jacobians of curves of genus 2 obtained by \textit{E. W. Howe} et al. [Ann. Inst. Fourier 59, No. 1, 239--289 (2009; Zbl 1236.11058)].
Reviewer: Dimitros Poulakis (Thessaloniki)Evaluation codes and their basic parameters.https://www.zbmath.org/1456.140342021-04-16T16:22:00+00:00"Jaramillo, Delio"https://www.zbmath.org/authors/?q=ai:jaramillo.delio"Vaz Pinto, Maria"https://www.zbmath.org/authors/?q=ai:vaz-pinto.maria"Villarreal, Rafael H."https://www.zbmath.org/authors/?q=ai:villarreal.rafael-heraclioLet \(K=GF(q)\) denote the finite field with \(q\) elements, where \(q\) is a
prime power, and let \(S=K[t_1,\dots,t_s]\) be the polynomial ring associated
with the affine space \({\mathbb{A}}^s\).
If \(X=\{P_1,\dots,P_m\}\subset {\mathbb{A}}^s\) is a
finite set of points then we can define a \(K\)-linear
evaluation map
\[
mathrm{ev}_X:S\to K^m, \ \ mathrm{ev}_X(f)=(f(P_1),\dots,f(P_m)).
\]
If \(L\subset S\) is a finite dimensional space then
its image under \(mathrm{ev}_X\), \(L_X=mathrm{ev}_X(L)\), is called an
\textit{evaluation code} on \(X\).
In the paper under review, the authors study the basic parameters of the
family of evaluation codes and those of certain interesting
subfamilies. The parameters investigated are
(a) the length \(|X|\),
(b) the dimension \(\dim_K(L_X)\),
(c) the generalized Hamming weights \(\delta_r(L_X)\), \(1\leq r \leq \dim_K(L_X)\).
The first main result of the authors is a formula for
\(\delta_r(L_X)\) in terms of ring-theoretic data involving
\(S\), the vanishing ideal of \(X\) and related ideals. The second
main result is a lower bound for the
\(\delta_r(L_X)\) in terms of what they call the footprint of \(L_X\), a
quantity expressed in terms of ring-theoretic data.
These two results are applied to
toric codes and to ``squarefree'' evaluation codes,
when \(X\) is taken to be the set of all \(K\)-rational
points of the affine torus.
For those families of codes, the authors obtain explicit formulas for both the
minimum distance (namely, \(\delta_1(L_X)\)) and the second
generalized Hamming weight \(\delta_2(L_X)\).
The exact statements are too technical to state in this review,
so the reader is referred to the paper for details.
Reviewer: David Joyner (Annapolis)Intersections between the norm-trace curve and some low degree curves.https://www.zbmath.org/1456.140322021-04-16T16:22:00+00:00"Bonini, Matteo"https://www.zbmath.org/authors/?q=ai:bonini.matteo"Sala, Massimiliano"https://www.zbmath.org/authors/?q=ai:sala.massimilianoThe affine plane curve \(\mathcal{X}\) defined over the finite field \(\mathbb{F}_{q^r}\) by the equation \[x^{\frac{q^r-1}{q-1}}= y^{q^{r-1}}+y^{q^{r-2}}+\cdots +y^q+y\] is called {\em norm-trace curve}. The norm-trace curve is a natural generalization of the Hermitian curve to any extension field \(\mathbb{F}_{q^r}\). The determination of the intersection of a given curve \(\mathcal{X}\) and low degree curves is often useful for the determination of the weight distribution of the AG code arising from \(\mathcal{X}\).
In this paper, the intersection of \(\mathcal{X}\) with cubics defined by equations of the form \(y = ax^3 + bx^2 + cx + d\), where \(a, b, c, d \in \mathbb{F}_{q^3}\), is studied. Especially, the intersection between the norm-trace curve and parabolas is characterised, and tools are provided in order to get sharp bounds for the number of intersections in the other cases. For this purpose specific irreducible surfaces over finite fields are studied. Furthermore, the weight distribution of the corresponding one-point codes are obtained.
Reviewer: Dimitros Poulakis (Thessaloniki)On the scalar complexity of Chudnovsky\(^2\) multiplication algorithm in finite fields.https://www.zbmath.org/1456.112352021-04-16T16:22:00+00:00"Ballet, Stéphane"https://www.zbmath.org/authors/?q=ai:ballet.stephane"Bonnecaze, Alexis"https://www.zbmath.org/authors/?q=ai:bonnecaze.alexis"Dang, Thanh-Hung"https://www.zbmath.org/authors/?q=ai:dang.thanh-hungThe paper under review deals with the multiplicative complexity of multiplication in a finite field \(F_{q^n}\) which is the number of multiplications required to multiply in the \(F_q\)-vector space \(F_{q^n}\). The types of multiplications in \(F_q\) are the scalar multiplication and the bilinear one. The scalar multiplication is
the multiplication by a constant in \(F_q\). The bilinear multiplication is a multiplication that depends on the elements of \(F_{q^n}\) that are multiplied. D. V. and G. V. Chudnovsky, generalizing interpolation algorithms on the projective line over \(F_q\) to algebraic curves of higher genus over \(F_q\), provided a method which enabled to prove the linearity of the bilinear complexity of multiplication in finite extensions
of a finite field [\textit{D. V. Chudnovsky} and \textit{G. V. Chudnovsky}, J. Complexity 4, No. 4, 285--316 (1988; Zbl 0668.68040)]. This is the so-called Chudnovsky\(^2\) algorithm.
In this paper, a new method of construction with an objective to reduce the scalar complexity of Chudnovsky\(^2\) multiplication algorithms is proposed. An optimized basis representation of the Riemann-Roch space \(L(2D)\) is sought in order to minimize the number of scalar multiplications in the algorithm.
In particular, the Baum-Shokrollahi construction for multiplication in \(F_{256}/F_4\) based on the elliptic Fermat curve \(x^3 + y^3 = 1\) is improved.
For the entire collection see [Zbl 1428.68013].
Reviewer: Dimitros Poulakis (Thessaloniki)