Linear spaces of quadrics and new good codes
Andries E. Brouwer · Bulletin of the Belgian Mathematical Society - Simon Stevin · 1998
A conjecture of Mario de Boer about the weights occurring in a space of quadrics is proved.Some record-breaking codes are constructed.Let V be a vector space of dimension m over F q and consider the space F of all quadratic forms on V .Then dim TheoremFor 0 ≤ t ≤ 1 2 m there do exist linear subspaces F t of F such that (i) these subspaces form a chain:all nonzero quadrics in F t have rank at least 2t (indeed, the associated symmetric bilinear forms all have rank at least 2t), (iv) the nonzero hyperbolic quadrics in F t have rank at least 2t + 2, (v) if m is odd, then the elliptic quadrics in F t have rank at least 2t + 2, (vi) if m = 2t, then the nonzero quadrics in F t are all elliptic.Parts (i)-(iv),(vi) are due to Mario de Boer [1].Part (v) was conjectured by him.One may construct a linear code C from F (and C t from F t ), by fixing one representative x in each projective point (1-space) x in the projective space P V , and use evaluation to get for each quadratic formIts weight is the number of projective points outside the quadric defined by Q.Clearly, this code has word length |P V | and dimension dim F t .