Prime-qudit polynomial code (QPyC)[1]

Description

Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_p\) (with prime \(p>n\)) prime-qudit CSS code constructed using two Reed-Solomon codes over \(GF(p)=\mathbb{Z}_p\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(\mathbb{Z}_p\), and let \(g\) be a number satisfying \(0\leq k \leq g < n\). Then, define degree-\(g\) polynomials \begin{align} f_{\mu\cup c}\left(x\right)=\mu_{0}+\mu_{1}x+\cdots+\mu_{k-1}x^{k-1}+c_{k}x^{k}+\cdots+c_{g}x^{g}\,, \end{align} where the first \(k\) coefficients are indexed by the coefficient vector \(\mu\in\mathbb{Z}_p^{\times k}\), and the remaining coefficients are indexed by the vector \(c\in\mathbb{Z}_p^{\times (g+1-k)}\). Logical states, labeled by \(\mu\), are superpositions of canonical basis states whose \(i\)th bit is \(f_{\mu\cup c}\), evaluated at \(\alpha_i\) and summed over all possible vectors \(c\), \begin{align} |\overline{\mu}\rangle=\sum_{c\in\mathbb{Z}_{p}^{\times(g+1-k)}}|f_{\mu\cup c}(\alpha_{1}),|f_{\mu\cup c}(\alpha_{2}),\cdots,|f_{\mu\cup c}(\alpha_{n})\rangle. \end{align}

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Zoo code information

Internal code ID: polynomial

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Zoo Code ID: polynomial

Cite as:
“Prime-qudit polynomial code (QPyC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polynomial
BibTeX:
@incollection{eczoo_polynomial, title={Prime-qudit polynomial code (QPyC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/polynomial} }
Permanent link:
https://errorcorrectionzoo.org/c/polynomial

References

[1]
M. Grassl, W. Geiselmann, and T. Beth, “Quantum Reed—Solomon Codes”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 231 (1999). DOI; quant-ph/9910059

Cite as:

“Prime-qudit polynomial code (QPyC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polynomial

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits/polynomial.yml.