Codes defined by forms of degree 2 on hermitian surfaces and Sørensen s conjecture - PDF

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Codes defined by forms of degree 2 on hermitian surfaces and Sørensen s conjecture Frédéric A. B. Edoukou CNRS, Institut de Mathématiques de Luminy Luminy case Marseille Cedex 9 - France E.mail

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Codes defined by forms of degree 2 on hermitian surfaces and Sørensen s conjecture Frédéric A. B. Edoukou CNRS, Institut de Mathématiques de Luminy Luminy case Marseille Cedex 9 - France E.mail : June 14, 2005 Abstract We study the functional codes C h (X) defined by G. Lachaud in [10] where X P N is an algebraic projective variety of degree d and dimension m. When X is a hermitian surface in P G(3, q), Sørensen in [15], has conjectured for h t (where q = t 2 ) a result : #X Z(f) (F q ) h(t 3 + t 2 t) + t + 1 which should give the exact value of the minimum distance of the functional code C h (X). In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e h = 2), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight. Keywords: functional codes, hermitian surface, hermitian curve, quadric, Sørensen s conjecture, weight. Mathematics Subject Classification: 05B25, 11T71, 14J29 1 1 Introduction After the works of Goppa, in coding theory, the construction of errorcorrecting codes from algebraic curves is now classical. From the works of renowned scholars in algebraic geometry, such as Manin, Vladut [11], we are able to consider codes built from higher dimensional algebraic varieties. Some of such codes have already been studied. Among others, projective Reed-Muller codes have been studied by G. Lachaud [9], A. B. Sørensen [14] and Y. Aubry [1]; codes on hermitian surfaces have also been studied by Chakravarti [4], and Hirschfeld, Vladut, Tsfasman [8]. In [10] G. Lachaud has also study the functional codes on a projective variety X of P N of degree d and dimension m. Theses codes have similarities with the well know projective Reed-Muller codes defined over a projective variety X of P N. In this paper, what we have in mind is to study the functional codes C h (X) introduced by G. Lachaud in [10] in the particular case where X is the nondegenerate hermitain surface X : x t x t x t x t+1 3 = 0 in P G(3, q) and q = t 2. Indeed, the hermitian surfaces are interesting since they are maximal with respect to their cardinality and they have also a beautiful geometry. Even in the case of quadric surfaces, Y. Aubry [1] is not able to determine precisely the minimum distance of the fonctional codes C 2 (X), when the quadrics are non-degenerate (elliptic or hyperbolic) and 1-degenerate (cones). Sørensen [15] has also tried to study the functional codes C h (X), on the hermitian surface X; but he is not able to find the minimum distance, even in the case h = 2. His work was motivated by the paper [3] (of Chakravarti) which undertake the study of the code C 2 (X) over F 4 by a complete computer search. Thus, he conjectured a result, which will be the main purpose of our work. The paper has been organized as follows. First of all, we recall somme notations and definitions of functional codes. Secondly, we state the conjecture of Sørensen. Thirdly, by using the projective classification of quadrics in [7], the properties of the hermitian surface X and the projective classification of the hermitian plane curves in [7], we give a demonstration of the conjecture of Sørensen in the case h = 2. Finally we conclude our work by expressing more clearly the exact parameters of the functional codes C 2 (X), the geometrical structure of the minimum weight codewords and their number, the second weight and the also geometrical structure of codewords reaching this second weight. 2 2 Notations We denote by F q the field with q elements, where q = p a is a power of the prime number p. Let V = A N+1 be the affine space of dimension N + 1 over F q. We denote also by P = P(V ) = P G(N, q) = P N the corrresponding projective space of dimension equal to N. We denote also by P h (V, F q ) the vector space of forms (that is, of homogeneous polynomials) of degree h in V with coefficients in F q. If f P h (V, F q ), we denote by Z(f), the set of zeros of f. Let X P a subvariety of X P, we call X Z(f) a degree h section of X, and we denote by X Z(f) (F q ), the algebraic set associated to X Z(f). 3 Functional codes defined by forms on projective varieties The subject of this section is to recall the construction of the functional codes defined by forms on projective varieties as it has been done by G. Lachaud in [10]. Definition 3.1 Let X be a finite set, X = (P 1,..., P n ). Let F(X, F q ) the space of all maps from X to F q. F(X, F q ) is a vector space; let F F(X, F q ) a subspace. Let c be the map defined by c : F(X, F q ) F n q f c(f) = (f(p 1 ),..., f(p n )) We call functional code defined by F and X, and we denote it by C(X, F ), the image of the map c restricted to F. c F : F F n q f c F (f) = (f(p 1 ),..., f(p n )) C(X, F ) = Imc F The functional code we have defined has the following parameters lenghc(x, F ) = n, dim C(X, F ) = dim F dim ker C F 3 Definition 3.2 Let c(f) a codeword. We call coweight of the word c(f), and we denote cw(f), the number of zeros digits in c(f). cw(f) = #{P X f(p ) = 0} (3.2.1) The weight of the word c(f) which we denote by w(c(f)) is w(c(f)) = n cw(f) The minimal distance of the functional code C(X, F ) is distc(x, F ) = n max f F cw(f) (3.2.2) Definition 3.3 We denote by (x 0 :... : x N ) the homogeneous coordinates in the projective space P N. Let W i be the set of points with homogeneous coordinates (x 0 :... : x N ) P N such that x 0 = x 1 =... x i 1 = 0 and x i 0. The family {W i } 0 i N is clearly a partition of P N. Let ν be a map defined by: for (x 0 :... : x N ) W i. If f P h (V, F q ) and Q P N, then ν : P N V {0} (x 0 :... : x N ) (0,..., 0, 1, x i+1,..., x N ) f(νq) = f(x 0,..., x N ) x h i with Q W i Definition 3.4 Let X P N an algebraic set defined over F q and h q. The code defined by forms of degree h on X is the functional code C h (X) = C(X, F ) obtained by taking F as the space of restrictions to X of forms in P h (V, F q ). We have length C h (X) = #X(F q ) (3.4.1) When the map c is injective, we get ( ) N + h dim C h (X) = h (3.4.2) 4 From (3.2.1), we deduce that cw(f) = #X Z(f) (F q ) (3.4.3) From (3.4.3), we can conclude that calculating the weights of the code C h (X), comes down to the computation of the number of points of degree h sections of X. 4 Codes defined by forms on hermitian varieties in P G(3, q) We now restrict ourselves to functional codes defined in particular algebraic sets in P G(3, q), where q = t 2 and t is a prime power. Thus, we consider the hermitian surface X defined by : X : x t x t x t x t+1 3 = 0 We know from (3.4.1) that the length of this code is length C h (X) = #X(F q ). In [2](p.1179) Bose and Chakravarti showed that #X(F q ) = (t 2 + 1)(t 3 + 1) (4.1). If h t the map c is injective and from (3.4.2), we can say that ( ) 3 + h dim C h (X) = (4.2). h For an estimate on the minimum distance of this code, as usually, a more careful analysis is necessary. 4.1 Codes defined by linear forms In [2] we have the following result. Theorem 4.1 Let H P 3 be a plane { t if H is not tangent to X, #X H (F q ) = t 3 + t if H is tangent to X. If H is tangent to X, X H is a singular hermitian curve of rank 2 in P G(2, t 2 ). More precisely, H X is a set of t+1 lines passing through a common point. Thus we get #X H (F q ) = (t + 1)t = t 3 + t If H is not tangent to X, X H is a non-singular hermitian curve in P G(2, t 2 ). Thus, we get #X H (F q ) = t Remark 4.2 From the above theorem the conjecture is true in the case of codes defined by linear forms (h=1). In that case, the code C 1 (X) (noted C(X)) is a two-weight code of dimension 4 whose weights are t 5 and t 5 + t Codes defined by forms of degree two (quadrics) In the case h = 2 and t = 2, the minimum distance of this code has been found by Paul P. Spurr in 1986 in his master s project by using a complete computer search; thus a [45, 10, 22]-code over F 4 has been found. In 1991, Anders B. Sørensen in [15], in his Ph. D. Thesis, without any computer programs, only with the geometric properties of quadrics and those of the non-singular cubic in [5], found the minimum distance of this code defined over F 4. Sørensen has tried to generalize the problem and conjectured that for h t (and t any prime power) we get #X Z(f) (F q ) h(t 3 + t 2 t) + t + 1 G. Lachaud has also proved in [10], with the help of proposition 2.3 [10] that #X Z(f) (F q ) h(t 3 + t 2 + t + 1) Unfortunately his bound is larger than that of Sørensen. We will now resolve the problem of Sorensen, in the case h = 2, and t any prime power. 5 Resolution of Sørensen s conjecture in the case h = 2 and t any prime power We need to recall the notion of the rank of a quadric Q. Definition 5.1 Let T be an inversible linear transformation defined over P N. Let i(q) be the number of indeterminates appearing explicitly in Q. The rank r(q) of the quadric Q is defined by r(q) = min T i(q T ) where T ranges over all inversible linear transformations over F q. A quadric Q is said to be degenerate if r(q) N + 1. Otherwise the quadric is nondegenerate. We have a particular property of quadric in [6] p Proposition 5.2 A quadric Q is degenerate if and only it is singular. We recall also the classification of quadrics in P G(3, q) under P GL(4, q) given by J.W.P. Hirschfeld [7] in table 1; thus we will consider two cases depending on the degeneration of the quadrics. In the whole paragraph we will note s(t) = 2(t 3 + t 2 t) + t + 1 s 2 (t) = 2t 3 + t Table.1 Quadrics in PG(3,q) Rank Description Q repeated plane pair of distinct planes line (quadric) cone hyperbolic quadric elliptic quadric q 2 + q + 1 2q 2 + q + 1 q + 1 q 2 + q + 1 (q + 1) 2 q Q is a degenerate quadric Degenerate quadrics in P G(3, q) are quadrics with rank stricly less than 4. So we need to distinguish four cases Rank Q = 1 In this case Q is a repeated plane. If Q is tangent to X, from theorem 4.1, Q X is a singular hermitian curve of rang 2 in P G(2, t 2 ), and Q X = t 3 + t s 2 (t). If Q is not tangent to X, from [2], Q X is a non-singular hermitian curve in P G(2, t 2 ), and Q X = t s 2 (t) Rank Q = 2 and Q is a line In this case we have Q = q + 1 = t 2 + 1, so Q X t s 2 (t). 7 5.1.3 Rank Q = 2 and Q is a pair of distinct planes (i) each plane is not tangent to X. If we write Q = P 1 P 2, then P 1 X and P 2 X are non-singular hermitian curves in P G(2, t 2 ), such that P 1 X = P 2 X = t Hence Q X 2(t 3 + 1) s 2 (t). (ii) one plane is tangent and the second is not tangent to X. Let us note Q = P 1 P 2, with P 1 tangent to X and P 2 not tangent to X. The singular hermitian curve of rang 2, P 1 X is in fact a set of (t + 1) lines passing through a point P, see theorem 4.1. Let l = P 1 P 2, and suppose l not contained in X; since l P 1, we have l X = (l P 1 ) X = l (P 1 X), so that l X is a single point, or a set of t + 1 points since two distinct lines in the projective plane have exactly one common point. We know that Q X = P 1 X + P 2 X P 1 P 2 X (5.1.3.ii) Since P 1 X = t 3 +t 2 +1 and P 2 X = t 3 +1, we get Q X = 2t 3 +t 2 +1 = s 2 (t) in the case l X is a single point, or Q X = 2t 3 + t 2 t + 1 s 2 (t) in the case l X is a set of t + 1 points. If l is contained in X, then from (5.1.3.ii) we get Q X = 2t s 2 (t). (iii) each plane is tangent to X. Let us note Q = P 1 P 2, with P 1, and P 2 tangent to X. We know that P 1 X and P 2 X as singular hermitian curves in P G(2, t 2 ) are each one a set of t + 1 lines passing through respectively by a point P 1 and a point P 2. Let l = P 1 P 2, then l X is a set of t + 1 points or a line (in the case l is contained in X). Since P 1 X = P 2 X = (t + 1)t 2 + 1, from the relation (5.1.3.ii) we conclude that either Q X = s(t) when l X is a set of t + 1 points, or Q X = 2t 3 + t = s 2 (t) when l is contained in X Rank Q = 3 (Q is a cone) Here, Q = q 2 + q + 1 = q(q + 1) + 1 and Q consists of the points on q + 1 lines, passing through a vertex S. Thus we can consider two cases. (i) no line in the cone is contained in each X. Let us note Q = q+1 i=1 L i where L i is a line q+1 Q X = (X L i ) (5.1.4.i.1) i=1 8 We have also X L i t + 1, so from the relation (5.1.4.i.1) Q X (t + 1)(q + 1). Hence Q X t 3 + t 2 + t + 1 s 2 (t). (ii) at least one line of the cone is contained in X. In P G(3, q) the number of lines passing through a point is q 2 + q + 1 = t 4 + t From the study of the hermitian surface X in [2], we have the following result. Proposition 5.3 If C is any point in X, there pass exactly t + 1 generators (i.e lines lying in X) which constitute the intersection with X of the tangent plane at C. And through C, there pass t lines lying in the tangent plane at C, out of which t + 1 are generators. The remaining t 2 t lines through C, which lie in the tangent plane meet X only in the single point C (there are called tangents to X). Each of the t 4 lines passing through C and not contained in the tangent plane at C intersect X in exactly t + 1 points, one of which is C (such lines are called secants to X). From this proposition, on the hermitian surface X, we can assert first of all that the maximum number of generators the cone Q can contained is t + 1. Secondly, if we suppose that the remaining t 2 t lines of the cone passing through the vertex S are all secants to X, we get Q X (t + 1)t (t 2 t)t. And therefore, we have Q X 2t s 2 (t). 5.2 Q is a non-degenerate quadric Non-degenerate quadrics in P G(3, q) are quadrics with rank equal to 4. Thus, from table 1, p. 7 there are two different types of such quadrics : hyperbolic quadrics and elliptic quadrics Q is a hyperbolic quadric When Q is a hyperbolic quadric, we have Q = (q + 1) 2. Before giving the demonstration of the conjecture of Sørensen when Q is a hyperbolic quadric, we need to recall some properties of lines, hermitian surfaces and hyperbolic quadrics in P G(3, q). Lemma 5.4 [7] In PG(3,q), the number of lines meeting three skew lines is q For a proof of this lemma see [7] p.3. Definition 5.5 The set of transversals of three skew lines is called a regulus. So that a regulus consists of q + 1 skew lines, and if l 1, l 2 and l 3 are any three of them, it is denoted by R(l 1, l 2, l 3 ). We have the following theorem, for the proof see [7] p.23. Theorem 5.6 In PG(3,q), q 2, let {l 1, l 2, l 3, l 4 } and {l 1, l 2, l 3, l 4 } be two sets of four lines such that any two of the same set are skew and such that 15 of the 16 pairs {l i, l j }meet in a point. Then the last pair also does. The above theorem says that the lines meeting l 1, l 2 and l 3 meet all the lines of R(l 1, l 2, l 3 ) = R and form a regulus, called the complementary regulus of R. From these results we have a good description of the hyperbolic quadric in P G(3, q). Theorem 5.7 The hyperbolic quadric Q consists of (q + 1) 2 points, which are all on a pair of complementary reguli. The two reguli are the two systems of generators of Q. A hyperbolic quadric whose complementary reguli are R and R is denoted by H(R, R ). In [7] p.123, we also have an important property on the intersection of a hyperbolic quadric and the hermitian surface X, in the case the hyperbolic quadric contained at least three lines of a regulus. Theorem 5.8 If Q = H(R, R ) has three skew lines on X, then X Q consists of 2(t + 1) lines of R 0 R 0 where R 0 R, R 0 R and R 0 = R 0 = t + 1. Remark 5.9 Theorem 5.8 says that, if a hyperbolic quadric contains three skew lines on the non-degenerate hermition surface X, then it contains exactly 2(t + 1) lines of the surface X, and t + 1 lines in each of the two reguli. So that, if X Q = C, then C = C 1 C 2 where C 1 and C 2 are respectively the set of simple points, and double points, C 2 = {l l l R 0, l R 0 }. We get C 1 = 2(t + 1)(t (t + 1)) and C 2 = (t + 1) 2. Hence X Q = C = 2t 3 + t We can consider only one regulus instead of working in the whole hyperbolic quadric. Thus, we are now in position to say that the apparently difficult problem on the intersection of an hyperbolic quadric and the hermitian surface is reduced to the four following simple cases. 10 (i)a regulus of the hyperbolic quadric H(R, R ) contains at least three skew lines on the surface X In this case, from the above remark we have X Q = s 2 (t). (ii)a regulus of the hyperbolic quadric H(R, R ) contains exactly two skew lines on the surface X We know that the hyperbolic quadric Q is generated by q+1 = t 2 +1 lines of a regulus R. The remaining t 2 1 lines of this regulus R which are not contained in the hermitian surface X, each of them, meet the hermitian surface X, in at most t + 1 points. Therefore we have X Q 2(t 2 + 1) + (t 2 1)(t + 1) = t 3 + 3t 2 t + 1 s 2 (t). (iii)a regulus of the hyperbolic quadric H(R, R ) contains exactly one line on the surface X We will suppose now that R is the regulus of the hyperbolic quadric which contains the only line of intersection. In this way, there are t 2 lines of R which are not contained in X. Each of them, meeting the hermitian surface in at most t + 1 points, therefore X Q t t 2 (t + 1) = t 3 + 2t s 2 (t). (iv) Each regulus of the hyperbolic quadric H(R, R ) does not contain any line on the surface X The hyperbolic quadric generated by one regulus for instance R, and each line of the regulus meeting the hermitian surface X in at most t+1 points, implies that X Q (t 2 +1)(t+1) = t 3 + t 2 + t + 1 s 2 (t), as in the case i) Q is an elliptic quadric When Q is an elliptic quadric, we have Q = q = t and no line is contained in Q (that is the characterization of an elliptic quadric in P G(3, q)). We need to distinguish two cases. (i) X Q =. In this case the problem is resolved. (ii) X Q =. Let us choose a point P X Q. We need the following proposition which can be found in [12]. Proposition 5.10 Let Q N be a non-degenerate quadric in P N and H a hyperplane of P N. Then: 11 If H is tangent to Q N, then Q N H is a degenerate quadric of rank N 1 in P N 1. If H is not tangent to Q N, then Q N H is a non-degenerate quadric in P N 1. Let H 1 be the tangent plane to the quadric at the point P. From the above proposition, we can deduce that Q H 1 is a degenerate quadric in P G(2, t 2 ) of rank equal to 2. In [6] p.156, table 7.2, J.W.P. Hirschfeld gave the classification of quadrics in P G(2, q). Thus, we get two cases: Q H 1 is either a pair of distinct lines or a point (a pair of conjuguate lines in P G(2, q) which meet in P G(2, t)). Since the quadric Q does not contain any line, we can say that Q H 1 is a point, Q H 1 = {P } and therefore Q X H 1 = {P } ( ) Let us take a line D passing through P and contained in the plane H 1. We consider also all the planes (H i ) i I passing through the line D. From the above proposition, if H i H 1, then Q H i is a non-degenerate quadric and therefore from proposition 5.2 a non-singular curve. On the other hand X H i is a hermitian curve (singular or not). If H i is not tangent to X, then X H i is a non-singular hermitian curve. Thus, X H i and Q H i are two non-singular projective plane curves, so they are irreductible. From the theorem of Bézout [13] chap.4, 2.1 p.236, we can say that X Q H i 2(t + 1) ( ) If H i is tangent to X, then X H i as a singular hermitian curve is from theorem 4.1 a set of t + 1 lines passing through a common point; each line meeting X Q in at most 2 points. Thus, we get We can now conclude that X Q H i 2(t + 1) ( ) if H i H 1, then X Q H i 2(t + 1) ( ) The point P belong to each X Q H i, and therefore from ( ) we get X Q H i {P } 2(t + 1) 1 ( ) There are exactly q + 1 planes passing through the line D, and their union generate the whole projective space P G(3, q). Thus, we get q+1 Q X = {P } (X Q H i {P }) ( ) i=2 12 From the relation ( ) we get q+1 Q X 1 + X Q H i {P } ( ) i=2 And from the relations ( ) and ( ) we deduce that Q X 1 + q(2(t + 1) 1). Finally we get, Q X 2t 3 + t = s 2 (t). It is now possible for us, to answer affirmatively to the conjecture of Sørensen in the case
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