Optical Coding Theory with Prime

can be obtained and eventually the probability of having i of the w pulse positions in the address codeword of the hard-limiting receiver being hit by K − 1 interferers is given by

$h_i^{\left(K-1\right)}=\frac{\left(2w\right)!}{\left(2i\right)!\left(2w-2i\right)!}{\displaystyle \sum _{j=0}^{2i}{\left(-1\right)}^{2i-j}\frac{\left(2i\right)!}{j!\left(2i-j\right)!}{p}_{j/2,j/2}^{K-1}}$

for i = {0,1/2,1, 3/2, …, w − 1/2, w}.

For a hard-limiting receiver, an error occurs when the received data bit is 0 but the address codeword has the number of pulse positions that are hit by interfering codewords being as high as the decision threshold Z_th. So, an error occurs only when there are as many as Z_th pulse positions in the address codeword seeing a sum of one hit after hard-limiting. This sum of one hit comes from the case when both sub-positions of a pulse position are hit with one-half-hit ...