Binary Goppa code: Difference between revisions

Revision as of 02:02, 31 May 2020

Revision as of 14:08, 13 June 2021

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==Construction and properties==

A binary Goppa code is defined by a [[polynomial]] <math>g(x)</math> of degree <math>t</math> over a [[finite field]] <math>GF(2^m)</math> withoutwith multipleno zerosrepeated roots, and a sequence <math>LL_1, ..., L_n</math> of <math>n</math> distinct elements from <math>GF(2^m)</math> that aren'tare not roots of the polynomial:<math>g</math>.

Codewords belong to the kernel of the syndrome function, forming a subspace of <math>\{0,1\}^n</math>:

: <math>\forall i,j \in \{0,\ldots,n-1\}: L_i \in GF(2^m) \land (L_i = L_j \implies i=j) \land g(L_i) \neq 0</math>

Codewords belong to the kernel of syndrome function, forming a subspace of <math>\{0,1\}^n</math>:

: <math>\Gamma(g,L)=\left\{ c \in \{0,1\}^n \left| \sum_{i=0}^{n-1} \frac{c_i}{x-L_i} \equiv 0 \mod g(x) \right. \right\}</math>