Golay Code Implementation – Encoding¶

The Golay code is encoded using modulo-2 division. Taking into account that the information bits are 12 per codeword, we must encode each 12-bit as one codeword.
The characteristic polynomials for the Golay code are:

• X11 + X9 + X7 + X6 + X5 + X + 1, coefficients AE3h.
• X11 + X10 + X6 + X5 + X4 + X2 + 1, coefficients C75h.

These polynomials generate different checkbits. For our encoding algorithm we will use the first polynomial AE3h.
Below, there is an example illustrating by hand the module-2 division process using exclusive-OR (XOR). The data is 555h. To generate 11 check-bit, we append 11 zero onto the bit-reversed data (LSB first).

Golay polynomial    info bits   zero fill                        
    |----------|  |----------||---------|                       
      AE3h            _555h(reversed)                      
     101011100011  10101010101000000000000   (11 zeros)                    
                   101011100011  (AE3h)                                
                   ------------ <---------- Exclusive-OR         
                        100100100000                             
                        101011100011                             
                        ------------                             
                          111100001100                           
                          101011100011                           
                          ------------                           
                           101111011110                          
                           101011100011                          
                           ------------                          
                              100111101000                       
                              101011100011                       
                              ------------                       
                               01100001011 <-- checkbits

So we have the 11-checkbits 01100001011, and then the bit-reversed remainder from the division 11010000110=686h.
After that, we put the codeword together for the transmission and we get 686555h which is called Systematic encoding (we can add a parity bit to the codeword to obtain an extended Golay).

The encoding algorithm is shown below. Long integers are used to conveniently store one codeword.

#define POLY 0xAE3 /* or use the other polynomial, 0xC75 */

unsigned long golay(unsigned long cw)

/* This function calculates [23,12] Golay codewords. 
     The format of the returned longint is 
     [checkbits(11),data(12)]. */

{
int i;
unsigned long c;
cw&=0xfffl;
c=cw; /* save original codeword /
for (i=1; i<=12; i++) / examine each data bit */ {
if (cw & 1) /* test data bit /
cw^=POLY; / XOR polynomial /
cw>>=1; / shift intermediate result /
}
return((cw<<12)|c); / assemble codeword */
}

The routine parity() adds the parity bit to complete the extended codeword. It is shown below.

int parity(unsigned long cw)
/* This function checks the overall parity of codeword cw.
If parity is even, 0 is returned, else 1. */

{
unsigned char p;

/* XOR the bytes of the codeword / 
  p=(unsigned char*)&cw; 
  p^=*((unsigned char*)&cw+1); 
  p^=*((unsigned char*)&cw+2);

/* XOR the halves of the intermediate result */ 
  p=p ^ (p>>4); 
  p=p ^ (p>>2); 
  p=p ^ (p>>1);

/* return the parity result */ 
  return(p & 1); 
}

References

http://aqdi.com/articles/using-the-golay-error-detection-and-correction-code-3/