Welcome to the Galois-field Kingdom.

Galois-field $$q$$-ary code Encodes $$K$$ states (codewords) in $$n$$ $$q$$-ary coordinates over the field $$GF(q)=\mathbb{F}_q$$ and has distance $$d$$. Usually denoted as $$(n,K,d)_q$$. The distance is the minimum number of coordinates where two strings in the code differ. Protection: Detects errors on up to $$d-1$$ coordinates, corrects erasure errors on up to $$d-1$$ coordinates, and corrects general errors on up to $$\left\lfloor (d-1)/2 \right\rfloor$$ coordinates. Parents: Error-correcting code (ECC). Parent of: Algebraic-geometry (AG) code.
Stub.
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 $$\alpha_i$$. Protection: Since each polynomial $$f_{\mu}$$ is of degree less than $$k$$, it can be determined from its values at $$k$$ points. This means that RS codes can correct erasures on up to $$n-k$$ registers. The resulting distance, $$d=n-k+1$$, saturates the Singleton bound. Parent of: Extended RS code, Folded RS code.
Residue AG code Also called a differential code. Stub. Parents: Algebraic-geometry (AG) code. Parent of: Goppa code. Cousins: Evaluation AG 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 $$R_a(z) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i}$$. Protection: The length $$n = |L|$$ , dimension $$k \geq n-mr$$ where $$r = \text{deg} G(z)$$, and the minimum distance $$d \geq r +1$$. Cousin of: Binary quantum Goppa code.
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. Parents: Reed-Solomon (RS) code.

## References

[1]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977
[2]
K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23, 426 (1952). DOI
[3]
I. S. Reed and G. Solomon, “Polynomial Codes Over Certain Finite Fields”, Journal of the Society for Industrial and Applied Mathematics 8, 300 (1960). DOI
[4]
R. C. Bose and D. K. Ray-Chaudhuri, “On a class of error correcting binary group codes”, Information and Control 3, 68 (1960). DOI
[5]
V. D. Goppa, "A new class of linear error-correcting codes", Probl. Peredach. Inform., vol. 6, no. 3, pp. 24-30, Sept. 1970.
[6]
V. D. Goppa, "Rational representation of codes and (Lg) codes", Probl. Peredach. Inform., vol. 7, no. 3, pp. 41-49, Sept. 1971.
[7]
E. Berlekamp, “Goppa codes”, IEEE Transactions on Information Theory 19, 590 (1973). DOI