Weight enumerator

MacWilliams identities

Let C ≤𝕂𝑛𝑞 be a [[Linear code|[𝑛,𝑘] code]] and C⟂ be its [𝑛,𝑛 −𝑘] Orthogonal code. Let 𝐴(𝑧) =wt⁡(C;𝑧) and 𝐵(𝑧) =wt⁡(C⟂;𝑧) be their respective weight enumerators. Then¹ code

𝐵(𝑧)=𝑞−1(1+(𝑞−1)𝑧𝑛)𝐴(1−𝑧1+(𝑞−1)𝑧)

Proof

proof

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Footnotes

1. 1999. Introduction to coding theory, §3.5, pp. 41–42 ↩