Codes from Veronese and Segre embeddings and Hamada's formula (Q5952333)

The authors investigate the properties of the code \(C_k^n(l,q)\) over \(\text{GF}(k)\) spanned by the characteristic functions of all hypersurfaces of degree \(l\) in \(\text{PG}(n,q)\). In particular, a monomial basis for any such code is explicitly determined, and a formula for the dimension is provided. Furthermore, it is shown that for any \(l\leq n\), the dimension of the code spanned by the characteristic functions of \(n-l\) flats of \(\text{PG}(n,q)\) is the same as the dimension of \(C_k^n(l,q)\), thus constructing an alternative to Hamada's formula. Finally, formulas for the dimension of the \(v\)th order generalised Reed-Muller code \(R_q(v,n+1)\) and the dimension of the code given by hyperplane intersections of Segre embeddings are deduced.