| Alternant code |
Stub. |
| Bose–Chaudhuri–Hocquenghem (BCH) code |
Stub. |
| Divisible code |
A linear \(q\)-ary code is \(\Delta\)-divisible if the Hamming weight of each of its codewords is divisible by \(\Delta\). A \(2\)-divisible (\(4\)-divisible) code is called even (doubly even) [1]. A code is called singly even if all codewords are even and at least one has weight equal to 2 modulo 4. |
| 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. |
| Folded RS code |
Stub. |
| Gabidulin code |
Also called a vector rank-metric code. A linear code over \(GF(q^N)\) that corrects errors over rank metric instead of the traditional Hamming distance. Every element \(GF(q^N)\) can be written as an \(N\)-dimensional vector with coefficients in \(GF(q)\), and the rank of a set of elements is rank of the matrix formed by their coefficients. |
| Generalized Reed-Solomon (GRS) code |
Stub. |
| Goppa code |
Let \( G(z) \) be a polynomial describing a projective plane curve with coefficients from \( GF(q^m) \) for some fixed integer \(m\). Let \( L \) be a finite subset of the extension field \( GF(q^m) \) where \(q\) is prime, meaning \( L = \{\alpha_1, \cdots, \alpha_n\} \) is a subset of nonzero elements of \( GF(q^m) \). A Goppa code \( \Gamma(L,G) \) is an \([n,k,d]\) linear code consisting of all vectors \(a = a_1, \cdots, a_n\) such that \( R_a(z) =0 \) modulo \(G(z)\), where \( R_a(z) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \). |
| Linear \(q\)-ary code |
An \((n,K,d)_q\) linear code is denoted as \([n,k,d]_q\), where \(k=\log_{q}K\) need not be an integer. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(\alpha x+ \beta y\) is also a codeword for any \(q\)-ary digits \(\alpha,\beta\). |
| 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. [2].
|
| 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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| Two-weight code |
Stub. |
| Wozencraft ensemble code |
Stub. |
| \(q\)-ary group code |
An \( [n,k]_{q} \) code based on a finite group \( G \) of size \(n \). A group code for an abelian group is called an abelian group code. |