| Extended RS code |
Stub. If \(f\in \mathcal{P}_k\) with \(k<q\), then \(\sum_{\alpha\in\mathbb{F}_q}f(\alpha)=0\) which implies RS codes are odd-like. Hence, by adding a parity check coordinate with evaluation point \(\alpha_0=0\) to an RS code on \(q-1\) registers, the distance increases to \(\hat{d}=d+1\). This addition yields an \([q,k,q-k+1]\) extended RS code. |
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| Folded RS code |
Stub. |
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| Maximum distance separable (MDS) code |
A \([n,k,d]_q\) \(q\)-ary linear code is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the Singleton bound
\begin{align}
d \leq n-k+1
\end{align}
becomes an equality. A code is called almost MDS (AMDS) when \(d=n-k\). A bound for general \(q\)-ary codes can also be formulated; see Thm. 1.9.10 in Ref. [1].
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| Maximum-rank distance (MRD) code |
Also called an optimal rank-distance code. An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the Singleton-like bound
\begin{align}
k \leq \max(n, m) (\min(n, m) - d + 1)
\end{align}
become an equality.
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MRD codes are matrix-code analogues of MDS codes. |
| Quantum maximum-distance-separable (MDS) code |
An \(((n,q^k,d))\) code constructed out of \(q\)-dimensional qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound
\begin{align}
2(d-1) \leq n-k
\end{align}
becomes an equality.
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| Reed-Solomon (RS) code |
An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(GF(q)\) with \(q>n\). An RS code encodes \(\mu=\{\mu_0,\cdots,\mu_{k-1}\}\) into \(\{f_\mu(\alpha_1),\cdots,f_\mu(\alpha_n)\}\), with polynomial
\begin{align}
f_\mu(x)=\mu_0+\mu_1 x + \cdots + \mu_{k-1}x^{k-1}.
\end{align}
In other words, each codeword \(\mu\) is a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\).
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Every RS code is MDS. If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [2]. |