MacWilliams identities

Let $C \leq 𝕂 𝑛 𝑞$ be a $[𝑛, 𝑘]$ code and $C ⟂$ be its $[𝑛, 𝑛 - 𝑘]$ Orthogonal code. Let $𝐴 (𝑧) = w t (C; 𝑧)$ and $𝐵 (𝑧) = w t (C ⟂; 𝑧)$ be their respective weight enumerators. Then¹ #m/thm/code

𝐵 (𝑧) = 𝑞 - 1 (1 + (𝑞 - 1) 𝑧 𝑛) 𝐴 (1 - 𝑧 1 + (𝑞 - 1) 𝑧)

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

1999. Introduction to coding theory, §3.5, pp. 41–42 ↩

MacWilliams identities

Footnotes