Optical Coding Theory with Prime

form the extended carrier-hopping prime codes with cardinality ${\left(p_1{p}_2\cdots p_{k-1}\right)}^2{p}_k+{\left(p_1{p}_2\cdots p_{k-2}\right)}^2{p}_{k-1}{p}_k+\cdots +p_1^2{p}_2{p}_3\cdots p_k+p_1{p}_2\cdots p_k$, L = p₁p₂⋯p_k wave-lengths, length N = p₁, and weight w ≤ p₁, where “⊙_pi” denotes a modulo-p_j mul-tiplication for j = {1,2,…,k} and “⊕p₁” denotes a modulo-p₁ addition.

Using k = 2, w = p₁ = 3, and p₂ = 5 as an example, this extended carrier-hopping prime code of L = p₁p₂ = 15 wavelengths and length N = p₁ = 3 has 45 codewords from group 1 and 15 codewords from group 2. The code-words in group 1 are denoted by x_i2,i1,l2,l1 = [(l₁,0),((i₁⊕₃,l₁) + 3i₂,1),(((2⊙₃i₁)⊕₃,l₁) + (2⊙₅i₂)3,2)] for i₁ ∈ [0,2], i₂ ∈ [1,4], l₁ ∈ [0,2], and l₂ = 0, and x_i2,i1,l2,l1 = [(l₁,0),((i₁⊕₃,l₁) + 3i₂,0),(((2⊙₃i₁)⊕₃l₁) + (2⊙₅i₂)3,0)] for i₁ ∈ [0,2], i