On the small weight codewords of the functional codes C_2(Q), Q a non-singular quadric

This linear code has length n = |Q| and dimension k = N + 2 2 -1. The third fundamental parameter of this linear code C 2 (Q) is its minimum distance d. We determine the 5 or 6 smallest weights of C 2 (Q) via geometrical arguments. Every homogeneous quadratic polynomial f in N + 1 variables defines a quadric Q ′ : f (X 0 , . . . , X N ) = 0. The small weight codewords of C 2 (Q) correspond to the quadrics of PG(N, q) having the largest intersections with Q.

We prove that these small weight codewords correspond to quadrics Q ′ which are the union of two hyperplanes of P G(N, q). Since there are different possibilities for the intersection of two hyperplanes with a non-singular quadric, we determine in this way the 5 or 6 smallest weights of the functional code C 2 (Q).

We also determine the exact number of codewords having the 5 or 6 smallest weights.

2 Quadrics in PG(N, q)

The non-singular quadrics in PG(N, q) are equal to:

• the non-singular parabolic quadrics Q(2N, q) in PG(2N, q) having standard equation

These quadrics contain q 2N -1 + • • •+ q + 1 points, and the largest dimensional spaces contained in a non-singular parabolic quadric of PG(2N, q) have dimension N -1,

• the non-singular hyperbolic quadrics Q + (2N +1, q) in PG(2N +1, q) having standard equation X 0 X 1 + • • • + X 2N X 2N +1 = 0. These quadrics contain (q N + 1)(q N +1 -1)/(q -1) = q 2N +q 2N -1 +• • •+q N +1 +2q N +q N -1 +• • •+q +1 points, and the largest dimensional spaces contained in a non-singular hyperbolic quadric of PG(2N + 1, q) have dimension N,

• the non-singular elliptic quadrics Q -(2N + 1, q) in PG(2N + 1, q) having standard equation f (X 0 , X 1 )+X 2 X 3 +• • •+X 2N X 2N +1 = 0, where f (X 0 , X 1 ) is an irreducible quadratic polynomial over F q . These quadrics contain (q N +1 + 1)(q N -1)/(q -1) = q 2N + q 2N -1 + • • • + q N +1 + q N -1 + • • • + q + 1 points, and the largest dimensional spaces contained in a non-singular elliptic quadric of PG(2N + 1, q) have dimension N -1.

All the quadrics of PG(N, q), including the non-singular quadrics, can be described as a quadric having an s-dimensional vertex π s of singular points, s ≥ -1, and having a non-singular base Q N -s-1 in an (N -s -1)-dimensional space skew to π s , denoted by π s Q N -s-1 .

We denote the largest dimensional spaces contained in a quadric by the generators of this quadric.

Since we will make heavily use of the sizes of (non-)singular quadrics of PG(N, q), we list these sizes explicitly.

• In PG(N, q), a quadric having an (N -2d -2)-dimensional vertex and a hyperbolic quadric Q + (2d + 1, q) as base has size

• In PG(N, q), a quadric having an (N -2d -2)-dimensional vertex and an elliptic quadric Q -(2d + 1, q) as base has size

• In PG(N, q), a quadric having an (N -2d -1)-dimensional vertex and a parabolic quadric Q(2d, q) as base has size

We note that the size of a (non-)singular quadric having a non-singular hyperbolic quadric as base, is always larger than the size of a (non-)singular quadric having a nonsingular parabolic quadric as base, which is itself always larger than the size of a (non-)singular quadric having a non-singular elliptic quadric as base.

The quadrics having the largest size are the union of two distinct hyperplanes of PG(N, q), and have size 2q N -1 + q N -2 + • • • + q + 1. The second largest quadrics in PG(N, q) are the quadrics having an (N -4)-dimensional vertex and a non-singular 3dimensional hyperbolic quadric Q + (3, q) as base. These quadrics have size q N -1 +2q N -2 + q N -3 + • • • + q + 1. The third largest quadrics in PG(N, q) have an (N -6)-dimensional vertex and a non-singular hyperbolic quadric Q + (5, q) as base. These quadrics have size

As we mentioned in the introduction, the smallest weight codewords of the code C 2 (Q) correspond to the largest intersections of Q with other quadrics Q ′ of PG(N, q). Let V be the intersection of the quadric Q with the quadric Q ′ . Two distinct quadrics Q and Q ′ define a unique pencil of quadrics λQ

then V also lies in every quadric λQ + µQ ′ of the pencil of quadrics defined by Q and Q ′ . A large intersection implies that there is a large quadric in the pencil. The q + 1 quadrics of the pencil contain altogether |PG(N, q)| + q|V | points, since the points of V lie in all the quadrics of the pencil and the other points of PG(N, q) lie in exactly one such quadric. So there is a quadric in the pencil containing at least (|PG(N, q)| + q|V |)/(q + 1) points.

If there is a quadric in the pencil which is equal to the union of two hyperplanes, then we ar

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