## Projective Code from Ovoid

In the projective space PG(3, b) with b > 2, a (b2 + 1)-cap known as ovoid or ovaloid exists. It can be defined as the set of all points (W : X : Y : Z) ∈ PG(3, b) such that

XY + Z2 + aZW + W2 = 0,

where aFb is such that the polynomial x2 + ax + 1 has no root in Fb. In PG(3, 2r) with r ≥ 3, r odd, there are also ovoids which are not quadrics, called Tits ovoids (see  and ).

The resulting orthogonal array is a linear OA(b4, b2 +1,Fb, 3), the dual code a linear [b2 +1, b2 − 3, 4]-code over Fb.

If b is a power of 2, a linear code with the same parameters can also be obtained as narrow-sense BCH-code C({1}) of length b2 +1 | b4 − 1. The fact that the minimum distance is at least 4 follows from the Roos-bound and considering the intervals {1, q} and { – (q + 1), 0} .

If the points of an ovoid are interpreted as the columns of the generator matrix of a projective code, a [b2 +1, 4, b2b]-code over Fb is obtained. All its non-zero code words have weight either b(b−1) or b2. The code words with weight b2 identify the affine subcaps with b2 points. Therefore a [b2 +1, 1, b2]-subcode exists, which can be used by construction X.

### Optimality

The ovoid is the largest possible cap in PG(3, b).