On the orthogonality of geometric codes (Q1431623)

The set \(C(n,q)\) of all functions from PG\((n,q)\) into GF\((q)\) is a vector space of dimension \(q^n+\cdots+q+1\) over GF\((q)\). The {geometric code} \(C_i(n,q)\) is the subspace of all functions \(f\in C\) such that \(\sum_{P\in\sigma}{f(P)}=0\), for any \(i\)-dimensional subspace \(\sigma\) of PG\((n,q)\). In this paper, a necessary and sufficient condition so that two geometric codes \(C_i\) and \(C_j\) with \(i\leq j\) are orthogonal is obtained, namely \(C_i\perp C_j\) in PG\((n,q)\), \(q=p^h\) if and only if \[ h<\log_p ((n+1-j)/i) +\log_p ((n+1-i)/j). \] This result is then applied to projective spaces PG\((n,q)\) of small dimension, and it is also shown that, in general, \(C_1\perp C_1\) if and only if \(n^2>q\) for \(q\) a square or \(n^2>q/p\) for \(q\) a non-square.