Codes, caps and linear spaces

Summary

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

<K> = <x⁽¹⁾,…, x^(k) > = V, (5)

<K> = <[x(¹⁾],…, [x^(k)] > = 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.