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arXiv:2509.07761

Dual of Algebraic Geometry codes from Hirzebruch surfaces

Barraud, Alix

Published Sep 9, 2025• 0 citations
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Abstract

In this paper, we give an explicit form for the dual of the algebraic geometry code $C_e(a,b)$ defined on an Hirzebruch surface $\mathcal{H}_e$ and parametrized by the divisor $aS_e + bF_e$, where $a,b\in\mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{Pic}( \mathcal{H}_e)$. Notably, we compute a lower bound for the minimum distance of $C_e(a,b)^\perp$. One of the main ingredient for our study is a new explicit form of the code $C_e(a,b)$ which we provide at the beginning of the paper. We also investigate some puncturing of $C_e(a,b)$, recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes $C_e(a,b)$, and construct CSS quantum codes from them.

DOI
10.48550/arxiv.2509.07761

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In this paper, we give an explicit form for the dual of the algebraic geometry code Ce(a,b)C_e(a,b)Ce​(a,b) defined on an Hirzebruch surface He\mathcal{H}_eHe​ and parametrized by the divisor aSe+bFeaS_e + bF_eaSe​+bFe​, where a,b∈Na,b\in\mathbb{N}a,b∈N and SeS_eSe​ and FeF_eFe​ generate the Picard group Pic(He)\mathrm{Pic}( \mathcal{H}_e)Pic(He​). Notably, we compute a lower bound for the minimum distance of Ce(a,b)⊥C_e(a,b)^\perpCe​(a,b)⊥. One of the main ingredient for our study is a new explicit form of the code Ce(a,b)C_e(a,b)Ce​(a,b) which we provide at the beginning of the paper. We also investigate some puncturing of Ce(a,b)C_e(a,b)Ce​(a,b), recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes Ce(a,b)C_e(a,b)Ce​(a,b), and construct CSS quantum codes from them.
Let $ \mathfrak{R}_{l, k} = {\mathbb F}_{p^m}[u_1, u_2, \cdots, u_k]/ \langle u_{i}^{l} = u_{i}, u_iu_j = u_ju_i = 0 \rangle $, where $ p $ is a prime, $ l $ is a positive integer, $ (l-1)\mid(p-1) $ and $ 1\leq i, j\leq k $. First, we define a Gray map $ \phi_{l, k} $ from $ \mathfrak{R}_{l, k}^n $ to $ {\mathbb F}_{p^m}^{((l-1)k+1)n} $, and study its Gray image. Further, we study the algebraic structure of $ \sigma $-self-orthogonal and $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $, and give the necessary and sufficient conditions for $ \lambda $-constacyclic codes over $ \mathfrak{R}_{l, k} $ to satisfy $ \sigma $-self-orthogonal and $ \sigma $-dual-containing. Finally, we construct quantum codes from $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $ using the CSS construction or Hermitian construction and compare new codes our obtained better than the existing codes in some recent references.
In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.
Let Fq\mathbb{F}_qFq​ be a finite field of qqq elements, for some prime power qqq, and let GGG be a finite group. A (left) group code, or simply a GGG-code, is a (left) ideal of the group algebra Fq[G]\mathbb{F}_q[G]Fq​[G]. In this paper, we provide a complete group-algebraic description for the Hermitian dual code of any DnD_nDn​-code over Fq2\mathbb{F}_{q^2}Fq2​, where DnD_nDn​ is a dihedral group of order 2n2n2n with nnn not divisible by char⁡(Fq2)\operatorname{char}(\mathbb{F}_{q^2})char(Fq2​), through a suitable Wedderburn-Artin decomposition of the group algebra Fq2[Dn]\mathbb{F}_{q^2}[D_n]Fq2​[Dn​], and we determine all distinct Hermitian self-orthogonal DnD_nDn​-codes over Fq2\mathbb{F}_{q^2}Fq2​. We also present a thorough representation of the Euclidean dual code of any QnQ_nQn​-code over Fq\mathbb{F}_qFq​, where QnQ_nQn​ is a generalised quaternion group of order 4n4n4n not divisible by char⁡(Fq)\operatorname{char}(\mathbb{F}_q)char(Fq​), via the Wedderburn-Artin decomposition of the group algebra Fq[Qn]\mathbb{F}_q[Q_n]Fq​[Qn​]. In particular, since the semisimple group algebras Fq2[Qn]\mathbb{F}_{q^2}[Q_n]Fq2​[Qn​] and Fq2[D2n]\mathbb{F}_{q^2}[D_{2n}]Fq2​[D2n​] are isomorphic, then the Hermitian dual code of any QnQ_nQn​-code has also been fully described. As an application of the Hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact, with this methodical approach, we recover some already known quantum codes that achieve the best known minimum distance for their length and dimension.
This work develops a geometric framework for constructing quantum error-correcting codes from weighted projective and orbifold structures, integrating algebraic geometry, divisor theory, and the CSS stabilizer formalism. Beginning with weighted projective spaces and their associated height and defect structures, the study builds classical AG-codes via evaluation on divisors adapted to orbifold singularities. These classical codes are lifted to quantum codes using self-orthogonality conditions and homological constructions, yielding a class of Quantum Weighted Algebraic Geometric (QWAG) codes. A central contribution is the formulation of a refined Singleton-type bound motivated by orbifold defect terms and effective genus corrections. While the classical quantum Singleton bound is recovered in the smooth case, the orbifold setting suggests additional geometric contributions that may adjust the theoretical distance bound. The refined bound is presented with partial justification under specific geometric hypotheses and framed as a conjectural extension in full generality. The monograph further provides explicit constructions, computational implementations in Sage/Python, and illustrative examples demonstrating how weighted geometry influences code parameters. This work establishes a structured bridge between orbifold geometry and quantum coding theory, outlining both concrete constructions and open problems for further mathematical development.
There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the projective lines, we construct new families of MDS quantum codes with parameters [[N,N−2K,K+1]]q[[N,N-2K,K+1]]_q[[N,N−2K,K+1]]q​ under the following conditions: i) N=t(q−1)+1N=t(q-1)+1N=t(q−1)+1 or t(q−1)+2t(q-1)+2t(q−1)+2 with t∣(q+1)t|(q+1)t∣(q+1) and K=⌊t(q−1)+12t⌋+1K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1K=⌊2tt(q−1)+1​⌋+1; ii) (n−1)∣(q2−1)(n-1)|(q^2-1)(n−1)∣(q2−1), N=nN=nN=n or N=n+1N=n+1N=n+1, K0=⌊n+q−1q+1⌋K_0=\lfloor\frac{n+q-1}{q+1}\rfloorK0​=⌊q+1n+q−1​⌋, and K≥K0+1K\ge K_0+1K≥K0​+1; iii) N=tq+1N=tq+1N=tq+1, ∀ 1≤t≤q\forall~1\le t\le q∀ 1≤t≤q and K=⌊tq+q−1q+1⌋+1K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1K=⌊q+1tq+q−1​⌋+1; iv) n∣(q2−1)n|(q^2-1)n∣(q2−1), n2=ngcd⁡(n,q+1)n_2=\frac{n}{\gcd (n,q+1)}n2​=gcd(n,q+1)n​, ∀ 1≤t≤q−1n2−1\forall~ 1\le t\le \frac{q-1}{n_2}-1∀ 1≤t≤n2​q−1​−1, N=(t+1)n+2N=(t+1)n+2N=(t+1)n+2 and K=⌊(t+1)n+1+q−1q+1⌋+1K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1K=⌊q+1(t+1)n+1+q−1​⌋+1.