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h1. 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 encodein (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); 

}