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AIMS Mathematics

Hermitian self-orthogonal infinitesimal evaluation codes over $ \mathbb{F}_{q^2}+u\mathbb{F}_{q^2} $ and applications to quantum codes

Sami Alabiad

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension $ R = \mathbb{F}_{q^2}+u \mathbb{F}_{q^2}, u^2 = 0, $ obtained by evaluating polynomials at perturbed points $ a_i+ub_i $. This evaluation produces a coupled value–derivative structure through the identity $ f(a_i+ub_i) = f_0(a_i)+u\bigl(b_i f_0'(a_i)+f_1(a_i)\bigr), $ which enriches classical evaluation codes with first-order infinitesimal corrections. We established the Hermitian duality theory for these codes and showed that Hermitian orthogonality over $ R $ decomposes into a residue-layer condition over $ \mathbb{F}_{q^2} $ together with a correction equation involving the infinitesimal parameters. This yields explicit criteria for Hermitian self-orthogonality. Using these criteria, we constructed several families of Hermitian self-orthogonal infinitesimal evaluation codes, including multiplier perturbation, locator perturbation, and subgroup–coset constructions. Via the Gray map and the Hermitian construction, these codes produce new families of $ q $-ary quantum stabilizer codes.

DOI
10.3934/math.2026694

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In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension R=Fq2+uFq2,u2=0,R = \mathbb{F}_{q^2}+u \mathbb{F}_{q^2}, u^2 = 0,R=Fq2​+uFq2​,u2=0, obtained by evaluating polynomials at perturbed points ai+ubia_i+ub_iai​+ubi​. This evaluation produces a coupled value–derivative structure through the identity f(ai+ubi)=f0(ai)+u(bif0′(ai)+f1(ai)),f(a_i+ub_i) = f_0(a_i)+u\bigl(b_i f_0'(a_i)+f_1(a_i)\bigr),f(ai​+ubi​)=f0​(ai​)+u(bi​f0′​(ai​)+f1​(ai​)), which enriches classical evaluation codes with first-order infinitesimal corrections. We established the Hermitian duality theory for these codes and showed that Hermitian orthogonality over RRR decomposes into a residue-layer condition over Fq2\mathbb{F}_{q^2}Fq2​ together with a correction equation involving the infinitesimal parameters. This yields explicit criteria for Hermitian self-orthogonality. Using these criteria, we constructed several families of Hermitian self-orthogonal infinitesimal evaluation codes, including multiplier perturbation, locator perturbation, and subgroup–coset constructions. Via the Gray map and the Hermitian construction, these codes produce new families of qqq-ary quantum stabilizer codes.
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.
Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for q>3q > 3q>3, we show that the code CD\mathcal{C}_DCD​ is self-orthogonal whenever the defining set DDD is GGG-invariant, where G⊆Fq∗G \subseteq \mathbb{F}_q^*G⊆Fq∗​ and ∣G∣>2|G| > 2∣G∣>2. For q=2,3q = 2, 3q=2,3, we characterize the self-orthogonality of CD\mathcal{C}_DCD​ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup G⊆Fq∗G \subseteq \mathbb{F}_q^*G⊆Fq∗​ with ∣G∣>2|G| > 2∣G∣>2 on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and sss-plateaued functions.
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We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F4which are records at http://codetables.de/.