On the geometry of small weight codewords of dual algebraic geometric codes

We investigate the geometry of the support of small weight codewords of dual algebraic geometric codes on smooth complete intersections by applying the powerful tools recently developed by Alain Couvreur. In particular, by restricting ourselves to the case of Hermitian codes, we recover and extend previous results obtained by the second named author joint with Marco Pellegrini and Massimiliano Sala.


Introduction
In the recent contribution [3], the number of small weight codewords for some families of Hermitian codes is determined.Besides explicit computation, the main ingredient in [3] is a nice geometric characterization of the points in the support of a minimum weight codeword, which turn out to be collinear (see Corollary 1 and Proposition 2 in [3]).
Here we show that such a property is not peculiar to Hermitian codes, but it holds in full generality for dual algebraic geometric codes on any smooth complete intersection projective variety of arbitrary dimension.Namely, by exploiting the algebro-geometric tools provided by [1], we prove the following: Theorem 1.Let X ⊂ P r , r ≥ 2, be a smooth connected complete intersection defined over F q .Let G be a divisor on X such that L(G) ⊇ H 0 (P r , O P r (m)).Let P 1 , . . ., P n be distinct rational points on X avoiding the support of G and let D := P 1 + . . .+ P n .Let d be the minimum distance of the code C(D, G) * and let {P i 1 , . . ., P i d } be the points in the support of a minimum weight codeword.(ii) If d ≤ 2m + 2 and no m + 2 of the P i j 's are collinear, then d = 2m + 2 and all the 2m + 2 points P i j lie on a plane conic.
(iii) If d ≤ 3m, no m + 2 of the P i j 's are collinear and no 2m + 2 of them lie on a plane conic, then d = 3m and all the 3m points P i j lie at the intersection of two complanar plane curves of respective degrees 3 and m.
If we focus on the explicit interesting example of Hermitian codes (as presented for instance in [2]), then our general result specializes as follows: Corollary 1.Let X ⊂ P 2 be the Hermitian curve of affine equation x q+1 = y q + y defined over F q 2 .Let G = P ρ 0 , where P 0 is the point at infinity, D = P 1 + . . .+ P n with n = q 3 , and let C(D, G) * be the Hermitian code.Let d be the minimum distance and let {P i 1 , . . ., P i d } be the points in the support of a minimum weight codeword.
, then all the points P i j are collinear.
(i) If d ≤ q, then all the points P i j are collinear.
(ii) If d ≤ 2q − 2, then either q of the points P i j are collinear, or all the points P i j lie on a plane conic.
(iii) If d ≤ 3q − 6, then either q of the points P i j are collinear, or 2q − 2 of the points P i j lie on a plane conic, or all the points P i j lie at the intersection of two complanar plane curves of respective degrees 3 and q − 2.
(i) If d ≤ q + 1, then all the points P i j are collinear.
(ii) If d ≤ 2q, then either q + 1 of the points P i j are collinear, or all the points P i j lie on a plane conic.
(iii) If d ≤ 3q − 3, then either q + 1 of the points P i j are collinear, or 2q of the points P i j lie on a plane conic, or all the points P i j lie at the intersection of two complanar plane curves of respective degrees 3 and q − 1.
In the case of Hermitian codes we may even describe the geometry of small, even not minimum, weight codewords (notice that our technical assumption L(G) = H 0 (P 2 , O P 2 (d − 2)) is satisfied by the so-called corner codes according to the terminology of [3], Definition 2): Proposition 1.Let X ⊂ P 2 be the Hermitian curve of affine equation x q+1 = y q + y defined over F q 2 .Let G = P ρ 0 , where P 0 is the point at infinity, D = P 1 + . . .+ P n with n = q 3 , and let C(D, G) * be the Hermitian code.If 0 ≤ ρ ≤ q 2 − q − 2 and L(G) = H 0 (P 2 , O P 2 (d − 2)), then at least d − 1 of the points {P i 1 , . . ., P i d+a } in the support of a codeword of weight d + a, where 0 ≤ a ≤ d − 3, are collinear.
In the next future, the second named author is going to apply the above geometric characterization to compute the number of small weight codewords for Hermitian corner codes, in the spirit of [3].Indeed, the present research is part of her PhD program, supervised by Massimiliano Sala.Both authors are grateful to him for strongly stimulating this joint project.The first named author has been partially supported by GNSAGA of INdAM and MIUR Cofin 2008 -Geometria delle varietà algebriche e dei loro spazi di moduli (Italy).

The proofs
Proof of Theorem 1.Let c ∈ C(D, G) * be a minimum weight codeword having support {P i 1 , . . ., P is } with s = d.By the definition of a dual code, we have s j=1 c j f (P i j ) = 0 for every f ∈ L(G) ⊇ H 0 (P r , O P r (m)).In particular, we have s j=1 c j ev P i j (f ) = 0 for every f ∈ H 0 (P r , O P r (m)).Hence ev P i j turn out to be linearly dependent in H 0 (P r , O P r (m)) * and, in the terminology of [1], Definition 2.4 and Definition 2.8, P i 1 , . . ., P is are m-linked.Now, we have d = s = m+2 in case (i) by [1], Proposition 6.1, d = s = 2m + 2 in case (ii) by [1], Proposition 7.2, and d = s = 3m in case (iii) by [1], Proposition 8.1.Finally, we complete case (i) by [1], Proposition 6.2, case (ii) by [1], Proposition 7.4, and case (iii) by [1], Proposition 9.1.
Proof of Proposition 1.If c j are the non-zero components of the corresponding codeword, then If they are not minimally (d − 2)-linked, then (up to reordering) we have (i) If d ≤ m + 2, then d = m + 2 and all the m + 2 points P i j are collinear in P r .

d+a− 1 j=1b
j f (P i j ) = 0 for every f ∈ H 0 (P 2 , O P 2 (d−2)).Our assumption L(G) = H 0 (P 2 , O P 2 (d−2)) implies that the b j 's are the components of a codeword of weight strictly less than d + a.By induction on a starting from Corollary 1 we conclude that at least d − 1 of them are collinear.Assume now that the points {P i 1 , . . ., P i d+a } are minimally (d−2)-linked.If they are not collinear, there exists a hyperplane H containing exactly l of them, with 2 ≤ l ≤ d + a − 1. By[1], Lemma 5.1, the remaining d + a − l points are (minimally) (d − 3)-linked and from[1], Proposition 7.2, it follows that at least (d − 1) of them are collinear since by our numerical assumption on a we have d + a − l ≤ 2(d − 3) + 1.