M17_spec/pages/04.appendix/04.golay-encoder/docs.md

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Golay Encoder	category docs

Extended Golay(24,12) code

The extended Golay(24,12) encoder uses generating polynomial g(x) given below to generate the 11 check bits. The check bits and an additional parity bit are appended to the 12 bit data, resulting in a 24 bit codeword. The resulting code is systematic, meaning that the input data (message) is embedded in the codeword.

g(x) = x^{11} + x^{10} + x^6 + x^5 + x^4 + x^2 + 1

This is equivalent to 0xC75 in hexadecimal notation. Both the generating matrix G and parity check matrix H are shown below.

( \begin{align} G = [I_{12}|P] = \left[ \begin{array}{cr} I_{12} \begin{matrix} 1&1&0&0&0&1&1&1&0&1&0&1\ 0&1&1&0&0&0&1&1&1&0&1&1\ 1&1&1&1&0&1&1&0&1&0&0&0\ 0&1&1&1&1&0&1&1&0&1&0&0\ 0&0&1&1&1&1&0&1&1&0&1&0\ 1&1&0&1&1&0&0&1&1&0&0&1\ 0&1&1&0&1&1&0&0&1&1&0&1\ 0&0&1&1&0&1&1&0&0&1&1&1\ 1&1&0&1&1&1&0&0&0&1&1&0\ 1&0&1&0&1&0&0&1&0&1&1&1\ 1&0&0&1&0&0&1&1&1&1&1&0\ 1&0&0&0&1&1&1&0&1&0&1&1 \end{matrix} \end{array} \right] \newline\newline H = [P^T|I_{12}] = \left[ \begin{array}{cr} \begin{matrix} 1&0&1&0&0&1&0&0&1&1&1&1\ 1&1&1&1&0&1&1&0&1&0&0&0\ 0&1&1&1&1&0&1&1&0&1&0&0\ 0&0&1&1&1&1&0&1&1&0&1&0\ 0&0&0&1&1&1&1&0&1&1&0&1\ 1&0&1&0&1&0&1&1&1&0&0&1\ 1&1&1&1&0&0&0&1&0&0&1&1\ 1&1&0&1&1&1&0&0&0&1&1&0\ 0&1&1&0&1&1&1&0&0&0&1&1\ 1&0&0&1&0&0&1&1&1&1&1&0\ 0&1&0&0&1&0&0&1&1&1&1&1\ 1&1&0&0&0&1&1&1&0&1&0&1 \end{matrix} I_{12} \end{array} \right] \end{align} )

The output of the Golay encoder is shown in the table below.

Table 1 Golay encoder details Field | Data | Check bits | Parity ----- | ---- | ---------- | ------ Position | 23..12 | 11..1 | 0 (LSB) Length | 12 | 11 | 1

Four of these 24-bit blocks are used to reconstruct the LSF.

Sample MATLAB/Octave code snippet for generating G and H matrices is shown below.

P = hex2poly('0xC75');
[H,G] = cyclgen(23, P);

G_P = G(1:12, 1:11);
I_K = eye(12);
G = [I_K G_P P.'];
H = [transpose([G_P P.']) I_K];

Issues to address...

• More details on Golay choice/performance
• Golay(24,12) matrix in C form