Talk:Binary Goppa code

Untitled
The parity check matrix $$H$$ is in the form $$H=VD$$. As currently written in the Wiki, matrix $$V$$ is $$(t+1)\times n$$, and $$D$$ is $$n \times n$$. Therefore, $$H$$ must be $$(t+1)\times n$$, whereas the text mentions that $$H$$ is a $$t$$-by-$$n$$ matrix. I believe matrix $$V$$ must be corrected. MSDousti (talk) 20:02, 16 July 2013 (UTC)
 * I have corrected the matrix by changing the highest powers from incorrect $$t$$ to correct $$t-1$$. Stefkar (talk) 18:39, 15 February 2018 (UTC)

Misleading use of $$\mathbb{Z}_n$$
Please change the line
 * $$\forall i,j \in \mathbb{Z}_n: L_i \in GF(2^m) \land L_i \neq L_j \land g(L_i) \neq 0$$

into
 * $$\forall i,j \in \lbrace 1,...,n \rbrace : L_i \in GF(2^m) \land L_i \neq L_j \land g(L_i) \neq 0$$

In the latter $$i,j$$ are just indizes running from $$1$$ through $$n$$, in the former they are elements of the coset ring $$\mathbb{Z}_n =\mathbb{Z}/n\mathbb{Z}$$, which has a lot more algebraic structure. (Addtion, multiplication and $$x = x + kn$$.) It was taking me nearly 15 minutes wondering what the purpose of the coset ring is and how it is related to $$n$$ before I understood that $$\mathbb{Z}_n$$ is meant to be an ugly short form for $$\lbrace 1,...,n \rbrace$$

Goppa Codes in Niederreiter's cryptosystem?
The text currently states: ... the binary Goppa codes are used in several post-quantum cryptosystems, notably McEliece cryptosystem and Niederreiter cryptosystem.

However, the Niederreiter cryptosystem proposed using Generalized Reed-Solomon (GRS) codes instead of Goppa codes in an attempt to make his cryptosystem more efficient and practical compared to McEliece's cryptosystem. - Markovisch (talk) 19:58, 19 April 2017 (UTC)

Hello! The GRS Niederreiter is certainly "faster" but it was broken by Sidelnikov&Shestakov (see ). As far as I know, binary Goppa codes and MDPCs are now the "simplest" codes useful in Niederreiter. Also see the thesis of Weger (2017) 212.79.106.136 (talk) 08:57, 3 December 2019 (UTC)