Codes, caps and linear spaces
Pier Vittorio Ceccherini, Giovanni Tallini · Cambridge University Press eBooks · 1981
CAPS OF PG(r,q) AND LINEAR CODES NOTATION Let V = V r+1,q be the (r+1)-dimensional vector space over the Galois field GF(q) and let S = S r,q = PG(r,q) be the related projective space of dimension r. If x є V \ {0}, then we denote by [x] the point of S related to x. Let us denote by the same symbol K the following: K = (x (1) , x (2) ,…,x (k) ) (ordered k-set of V), (1) K = ([x (1) ], [x (2) ],…,[x (k) ]) (ordered k-set of S), (2) K = [x (1) , x (2) ,…,x (k) ] ((r+1)xk matrix over GF(q)), (3) where x (1) , x (2) ,…,x (k) , are (column) vectors pairwise independent and spanning V. The latter condition implies r + 1 ≤ k, (4) and we have = = V, (5) = = S, (6) rank K = r + 1. (7) CODES AND ORDERED SETS OF POINTS With K as above, let C = C(K) be the linear code of V k,q defined by C(K) = {x є V k,q | Kx = 0}. By (7) we have that dim C(K) = k - (r+1). Moreover, in order that each column of the matrix K is a non-zero vector, the code C(K) satisfies the following condition: (C) The code C does not contain any basis vector, i.e. it does not contain any fundamental subspace.