EA mixed-alphabet Reed-Solomon c-q code

EA mixed-alphabet Reed-Solomon c-q code[1]

Alternative names: Mixed-alphabet Reed-Solomon EACC code, Mixed-alphabet RS entanglement-assisted classical code.

Description

Entanglement-assisted c-q code obtained from a mixed-alphabet Reed-Solomon construction over \(\mathbb{F}_q\) and \(\mathbb{F}_{q^2}\). A codeword of an \([n,k,d;c]_q\) code consists of \(n-c\) symbols transmitted directly over \(q\)-dimensional quantum systems and \(c\) symbols transmitted through super-dense coding using \(c\) pre-shared maximally entangled qudit pairs.

More explicitly, the code evaluates all polynomials \(f\in\mathbb{F}_q[x]\) of degree at most \(k-1\) at \(n-c\) distinct points \(\alpha_i\in\mathbb{F}_q\) and \(c\) representatives \(\gamma_j\in\mathbb{F}_{q^2}\setminus\mathbb{F}_q\), choosing at most one element from each conjugate pair \(\{\gamma,\gamma^q\}\). This yields codewords in \(\mathbb{F}_q^{n-c}\times\mathbb{F}_{q^2}^{c}\), where each \(\mathbb{F}_{q^2}\) symbol is identified with two \(q\)-ary symbols for dense coding.

Protection

Let \(n_1=n-c\) and \(n_2=c\). If \(n_1\geq k-1\), then the minimum distance is \(d=n-k+1\), saturating the classical Singleton bound. If \(n_1<k-1\), then \begin{align} d=\left\lceil\frac{n-k+1+n_2}{2}\right\rceil~, \tag*{(1)}\end{align} which can exceed the classical Singleton bound because a known erasure of an \(\mathbb{F}_{q^2}\) position removes a two-symbol block [1].

Rate

The construction can have \(k>n\) when \(c>0\), since each dense-coded position carries two \(q\)-ary symbols. Its length is bounded by \(n\leq q+(q^2-q)/2=(q^2+q)/2\), with a possible one-symbol extension using the point at infinity [1]. In the range \(n\leq q+(q^2-q)/2\) and \(n-q\leq c\), the distance formula above meets the block-erasure bound of Ref. [1].

Cousins

Reed-Solomon (RS) code— EA mixed-alphabet RS c-q codes use Reed-Solomon polynomial evaluation, but evaluate over both \(\mathbb{F}_q\) and selected representatives from \(\mathbb{F}_{q^2}\setminus\mathbb{F}_q\) to support direct and dense-coded channel uses [1].
Maximum distance separable (MDS) code— EA mixed-alphabet RS c-q codes can saturate a block-erasure bound and, in some parameter ranges, exceed the classical Singleton bound for ordinary \(q\)-ary codes [1].

Member of code lists

Primary Hierarchy

Classical-quantum Domain

Entanglement-assisted (EA) c-q code

Parents

Entanglement-assisted (EA) c-q code

EA mixed-alphabet Reed-Solomon c-q code

References

[1]: T. Prasad and M. Grassl, “Codes for entanglement-assisted classical communication”, npj Quantum Information 11, (2025) arXiv:2310.19774 DOI

Page edit log

Victor V. Albert (2026-05-26) — most recent

Cite as:

“EA mixed-alphabet Reed-Solomon c-q code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ea_mixed_alphabet_reed_solomon

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical_into_quantum/ea_mixed_alphabet_reed_solomon.yml.