Question

If C is a subspace of $F^n_{2}$, prove that $dim(C) + dim(C^\perp) = n$ .

(C is a binary linear code with length n and dimension k, $C^\perp$ is the dual code of C)

Answer 1

le cis a subspace of the show linke) + dim (et) = n. Since oua subspace of 5n, then S i ... z o be the basis for a Then din c) =k. The corresponding matrix for c an Anxk la we know that, C- span { t ... n} = Column space of A - COLCA) since null(AT) = Col (A) – then Null (AT) - Colla)* = ct 'dim Cct) = dim (N(AT)) = nullity aut CAT) Now, where Here Rank (AT) = + Nullity (AT) = n (by sank nullity theoreem) - 1. Let a vectore B= 13.5 Ton its Pow reeduced Echelon form

So nullsface of (AT) =n-ki Ince rank (A) = rank (AT), from eg we get sant (A) ant rank(A) + nullity CAT) = dim (Col(A)) + dim (Aule (AT))=n (as ten k+ n-K= n) din C+ dim ct = n.