Optical Coding Theory with Prime

Similarly, the one-hit probability of the address codeword originating from group l₁ = {1,2, …, p−1} when it correlates with an interfering codeword from any group is derived as

$\begin{array}{lll}{q}_{i,i^{\prime },j,j^{\prime }}^{\left(1\right)}\hfill & =\hfill & \frac{1}{2}[\frac{1}{N_{j^{\prime }}}\times \frac{\min \left\{w_i,w_j\right\}\left(p-1\right)}{{\beta }_{j^{\prime }}-1}+\frac{w_i{w}_j-\min \left\{w_i,w_j\right\}}{N_{j^{\prime }}}\hfill \\ \times \frac{1}{{\beta }_{j^{\prime }}-1}+\frac{w_i{w}_j-\min \left\{w_i,w_j\right\}}{N_{j^{\prime }}}\times \frac{\left(p-2\right)}{{\beta }_{j^{\prime }}{p}^2-1}\hfill \\ +\frac{w_i{w}_j-\min \left\{w_i,w_j\right\}}{N_{j^{\prime }}}\times \frac{1}{{\beta }_{j^{\prime }}-1}+\frac{w_i{w}_j}{N_{j^{\prime }}}\times \frac{\left[\left({\beta }_{j^{\prime }}{p}^2\right)-1\right]p}{{\beta }_{j^{\prime }}-1}]\hfill \end{array}$

The derivation follows the rationale used for $q_{i,i^{\prime },j,j^{\prime }}^{\left(0\right)}$. The first four product terms in the brackets account for the one-hit probabilities caused by an interfering codeword coming from any group l = {0,1, …, p − 1} and from the same time-spreading OOC codeword as that of the address codeword. The first product term accounts for the one-hit probability caused by the interfering codeword aligning with the address ...