Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum error-correction, the observed presence of asymmetry in some quantum channels led to the study of asymmetric quantum codes (AQECCs) where we no longer assume that the different types of errors are equiprobable. This paper considers quantum codes constructed from the two-point codes. In the asymmetric case, we show strict improvements over all possible finite fields for a range of designed distances. We produce large dimension pure AQECC and small dimension impure AQECC that have better parameters than AQECC from one-point codes. Numerical results for the Hermitian curves over F16 and F64 are used to illustrate the gain.Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum error-correction, the observed presence of asymmetry in some quantum channels led to the study of asymmetric quantum codes (AQECCs) where we no longer assume that the different types of errors are equiprobable. This paper considers quantum codes constructed from the two-point codes. In the asymmetric case, we show strict improvements over all possible finite fields for a range of designed distances. We produce large dimension pure AQECC and small dimension impure AQECC that have better parameters than AQECC from one-point codes. Numerical results for the Hermitian curves over F16 and F64 are used to illustrate the gain.

arXiv:1102.3605Feb 17, 20111 citation

Nonbinary quantum codes derived from finite geometries

David Clark, Vladimir D. Tonchev

Finite Fields and Their ApplicationsJul 5, 20112 citations

Reinforcement learning decoding for Hermitian AG quantum stabilizer codes

Vahid Nourozi

Abstract We provide a corrected, consistent development of classical algebraic geometry (AG) codes and associated nonbinary quantum stabilizer codes built fromthe Hermitian curve $y^q+y=x^{q+1}$ over $\Fqsq$. This resolves earlier inconsistencies around the choice of $m$ , point counts, dual indices, and field of the inner product. The curve has genus $g=\frac{q(q-1)}{2}$ , $q^3{+}1$ $\Fqsq$-rational points, and a unique point at infinity; setting $D$ to be the sum of all affine $\Fqsq$-points gives length $n=\deg D=q^3$ . For one-point divisors $G=rP_\infty$ we give a monomial basis for $\mathcal{L}(G)$ , the standard duality formula $C_{\mathcal{L}}(D,rP_\infty)^\perp=C_{\mathcal{L}}\!\big(D,(n+2g-2-r)P_\infty\big)$ , and a simple sufficient condition for Hermitian self-orthogonality: $(q+1)r\le n+2g-2$ (equivalently, $r\le q^2-2$ ).Under this condition, Ashikhmin--Knill yields $q$ -ary stabilizer codes $[[n,n-2k,\ge d_H^\perp]]_q$ with $k=\dim C_{\mathcal{L}}(D,rP_\infty)$ and $d_H^\perp\ge r-2g+2$ .We also retain a complete reinforcement-learning (RL) section: an \emph{RL-on-Greedy} decoder that first applies a greedy pass then a Deep Q-Network to resolve residual syndromes. We include the original evaluation plot and all references, and clarify that the RL case study can be applied to any stabilizer code (including, but not limited to, those from Hermitian AG codes).Abstract We provide a corrected, consistent development of classical algebraic geometry (AG) codes and associated nonbinary quantum stabilizer codes built fromthe Hermitian curve $y^q+y=x^{q+1}$ over $\Fqsq$. This resolves earlier inconsistencies around the choice of $m$, point counts, dual indices, and field of the inner product. The curve has genus $g=\frac{q(q-1)}{2}$, $q^3{+}1$ $\Fqsq$-rational points, and a unique point at infinity; setting $D$ to be the sum of all affine $\Fqsq$-points gives length $n=\deg D=q^3$. For one-point divisors $G=rP_\infty$ we give a monomial basis for $\mathcal{L}(G)$, the standard duality formula $C_{\mathcal{L}}(D,rP_\infty)^\perp=C_{\mathcal{L}}\!\big(D,(n+2g-2-r)P_\infty\big)$, and a simple sufficient condition for Hermitian self-orthogonality: $(q+1)r\le n+2g-2$ (equivalently, $r\le q^2-2$).Under this condition, Ashikhmin--Knill yields $q$-ary stabilizer codes $[[n,n-2k,\ge d_H^\perp]]_q$ with $k=\dim C_{\mathcal{L}}(D,rP_\infty)$ and $d_H^\perp\ge r-2g+2$.We also retain a complete reinforcement-learning (RL) section: an \emph{RL-on-Greedy} decoder that first applies a greedy pass then a Deep Q-Network to resolve residual syndromes. We include the original evaluation plot and all references, and clarify that the RL case study can be applied to any stabilizer code (including, but not limited to, those from Hermitian AG codes).

Oct 6, 20250 citations

Abstract We provide a corrected, consistent development of classical algebraic geometry (AG) codes and associated nonbinary quantum stabilizer codes built fromthe Hermitian curve

y^q+y=x^{q+1}

over $\Fqsq$. This resolves earlier inconsistencies around the choice of

m

, point counts, dual indices, and field of the inner product. The curve has genus

g=\frac{q(q-1)}{2}

q^3{+}1

$\Fqsq$-rational points, and a unique point at infinity; setting

D

to be the sum of all affine $\Fqsq$-points gives length

n=\deg D=q^3

. For one-point divisors

G=rP_\infty

we give a monomial basis for

\mathcal{L}(G)

, the standard duality formula

C_{\mathcal{L}}(D,rP_\infty)^\perp=C_{\mathcal{L}}\!\big(D,(n+2g-2-r)P_\infty\big)

, and a simple sufficient condition for Hermitian self-orthogonality:

(q+1)r\le n+2g-2

(equivalently,

r\le q^2-2

).Under this condition, Ashikhmin--Knill yields

q

-ary stabilizer codes

[[n,n-2k,\ge d_H^\perp]]_q

with

k=\dim C_{\mathcal{L}}(D,rP_\infty)

and

d_H^\perp\ge r-2g+2

.We also retain a complete reinforcement-learning (RL) section: an \emph{RL-on-Greedy} decoder that first applies a greedy pass then a Deep Q-Network to resolve residual syndromes. We include the original evaluation plot and all references, and clarify that the RL case study can be applied to any stabilizer code (including, but not limited to, those from Hermitian AG codes).