Essential Coding Theory

By elementary linear algebra involving row operations (replacing row i by row i plus row j), and column exchanges, we can write G as (Ik|A) where A is some k×n−k matrix. Now the matrix H = ( −A In−k ) satisfies GH = 0. Thus every vector y = xG satisfies yH = 0. Furthermore standard linear algebra implies that the space of vectors that satisfies yH = 0 is at most k and so this must be the space spanned by G. The relationship between n, k,m is thus m = n− k. (b) Give an efficient algorithm to compute such an H, given G, and vice versa.