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The Klein Quartic, the Fano Plane and Curves Representing Designs

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The Klein Quartic, the Fano Plane and Curves Representing Designs The Klein quartic, the Fano plane and curves representing designs Ruud Pellikaan ∗ Dedicated to the 60-th birthday of Richard E. Blahut, in Codes, Curves and Signals: Common Threads in Communications, (A. Vardy Ed.), pp. 9-20, Kluwer Acad. Publ., Dordrecht 1998. 1 Introduction The projective plane curve with defining equation X3Y + Y 3Z + Z3X = 0. has been studied for numerous reasons since Klein [19]. It was shown by Hurwitz [16, 17] that a curve considered over the complex numbers has at most 84(g − 1) automorphims, where g is the genus and g > 1. The above curve, nowadays called the Klein quartic, has genus 3 and an automorphism group of 168 elements. So it is optimal with respect to the number of automorphisms. This curve has 24 points with coordinates in the finite field of 8 elements. This is optimal with respect to Serre’s improvement of the Hasse-Weil bound: √ N ≤ q + 1 + gb2 qc, ∗Department of Mathematics and Computing Science, Technical University of Eind- hoven , P.O. Box 513, 5600 MB Eindhoven, The Netherlands. 1 where N is the number of Fq-rational points of the curve [22]. Therefore the geometric Goppa codes on the Klein quartic have good parameters, and these were studied by many authors after [13]. R.E. Blahut challenged coding theorists to give a selfcontained account of the properties of codes on curves [3]. In particular it should be possible to explain this for codes on the Klein quartic. With the decoding algorithm by a majority vote among unknown syndromes [7] his dream became true. The elementary treatment of algebraic geometry codes [8, 9, 10, 14] is now based on the observation that: if a decoding algorithm corrects t errors, then the minimum distance of the code is at least 2t + 1. This is very much the point of view of Blahut in his book [2] where he proves the BCH bound on the minimum distance as a corollary of a decoding algorithm. When Blahut was in Eindhoven for a series of lectures on algebraic ge- ometry codes [4], he gave the above picture of the Klein quartic over F8, see 3 Figure 1. Let α be an element of F8 such that α + α + 1 = 0. Then α is an primitive element of F8. The dot at place (i, j) in the diagram denotes that (αi, αj) is a point of the Klein quartic with affine equation: X3Y + Y 3 + X = 0. We get all points with nonzero coordinates in F8 in this way. It was noted by J.J. Seidel, who was in the audience, that this diagram can be interpreted as the incidence matrix of the the Fano plane 1, that is the projective plane of order two. This means that every row has three dots. The same holds for the columns. Furthermore for every two distinct columns there is exactly one row with a dot in that row and in those given columns. This paper grew out of an attempt to understand this phenomenon. Con- sider the affine plane curve Xq with equation q+1 q+1 Fq(X, Y ) = X Y + Y + X = 0 ∗ over the finite field Fq3 . Let Pq be the cyclic subgroup of the units Fq3 of order q2 + q + 1. So ∗ q2+q+1 Pq = { x ∈ Fq3 | x = 1 } 1This refers to G. Fano (1871-1952) one of the pioneers in finite geometry, not to be confused with R.M. Fano known from information theory 2 α6 • • • α5 • • • α4 • • • α3 • • • α2 • • • α1 • • • α0 • • • α0 α1 α2 α3 α4 α5 α6 ∗2 Figure 1: The Klein quartic in F8 Define an incidence relation on Pq by x is incident with y if and only if Fq(x, y) = 0. Then this defines an incidence structure of a projective plane of order q, that is to say a 2−(q2 +q +1, q +1, 1) design. This means that there are q2 +q +1 points, that the lines By = {x ∈ Pq | Fq(x, y) = 0 } have q + 1 elements for all y ∈ Pq, that there is exactly one line incident with two distinct points, and that there is exactly one point incident with two distinct lines. For every q a power of a prime there exists a projective plane of order q, called the the Desarguesian plane and denoted by PG(2, q). Not all projec- tive planes of a given order are isomorphic. It was explained to me by S. Ball and A. Blokhuis [1] that the above projective plane of order q is Desarguesian. In Section 2 the notion of curves or bivariate polynomials representing t- designs is defined. It is shown that Fq(X, Y ) represents PG(2, q). In Section 3 the number of Fq3 -rational points is computed of the projective curve Xq with affine equation Fq(X, Y ) = 0. In Section 4 a general construction is given of a bivariate polynomial representing symmetric designs, starting with a univariate polynomial. A complete characterization is given of polynomials αXnY + Y n+1 + βX that represent finite projective planes. It is shown that the design of the points/hyperplanes of the projective geometry PG(m, q) is represented by a curve. 3 2 Designs represented by curves The name design comes from the design of statistical experiments. For its basic properties and its relations with coding theory the interested reader is referred to [5, 20]. Definition 2.1 Let t, v, k and λ be positive integers. Let P be a set of v elements, called points. Let B be a collection of elements, called blocks. Let I ⊂ P × B be a relation between points and blocks. We say that P is incident with B if (P, B) ∈ I. Assume that every block B ∈ B has k elements. Suppose that for every choice of t distinct points of P there are exactly λ blocks of B that are incident with the given t-set. Then the triple (P, B, I) is called a t − (v, k, λ) design. v k Let b be the number of blocks of a t − (v, k, λ) design. Then b = λ t / t . Every point is contained in the same number r = bk/v of blocks. A 2-design is called symmetric if b = v. If (P, B, I) is a symmetric 2- design, then (B, P, I−1) is also a 2-design with the same parameters, where I−1 = {(B, P )|(P, B) ∈ I}. It is called the dual design. Definition 2.2 A 2 − (n2 + n + 1, n + 1, 1) design is called a projective plane of order n. The blocks are than called lines. For properties and constructions of finite projective planes one consults [6, 15]. The standard example is the Desarguesian projective plane PG(2, q) of order q, where q is a power of a prime. A point P of PG(2, q) is a 1- 3 dimensional linear subspace of Fq, a line B of PG(2, q) is a 2-dimensional 3 linear subspace of Fq, and P is incident with B if and only if P ⊂ B. Definition 2.3 Let F ∈ Fq[X, Y ]. Let P and B be subsets of Fq. Let I be the relation on P × B defined by (x, y) ∈ I if and only if F (x, y) = 0 for x ∈ P and y ∈ B. We say that (P, B,F ) or F represents a t − (v, k, λ) design if (P, B, I) is a t − (v, k, λ) design, degX (F ) = k and degY (F ) = r is the number of blocks through a given point of this design. 4 Consider the polynomial of the introduction q+1 q+1 Fq(X, Y ) = X Y + Y + X q2+q+1 over the field Fq3 . Define Pq = {x ∈ Fq3 | x = 1}. Let Iq be the inci- dence relation on Pq × Pq defined by (x, y) ∈ Iq if and only if Fq(x, y) = 0. The following Lemma will be used several times in this paper. q+1 Lemma 2.4 The polynomial fq(T ) = T + T + 1 has q + 1 distinct zeros in Pq. q+1 Proof. Let α + α + 1 = 0 for some α in the algebraic closure of Fq. Then raising this equation to the q-th power and multiplying with α gives αq2+q+1 + αq+1 + α = 0. So αq2+q+1 = −(αq+1 + α) = 1. Hence α ∈ Pq. The polynomial fq(T ) has q + 1 distinct zeros, since its 0 q derivative fq(T ) = T + 1 has greatest common divisor 1 with fq(T ). 2 2 Proposition 2.5 The triple (Pq, Pq,Fq) represents a 2−(q +q + 1, q + 1, 1) design. Proof. See also [1]. The polynomial fq(T ) has q + 1 distinct zeros in Pq by −q Lemma 2.4. Let x ∈ Pq and let α be a zero of fq(T ). Define y = αx . Then q y ∈ Pq. Substituting α = x y in fq(T ) gives (xqy)q+1 + xqy + 1 = 0 Multiplying with x gives yq+1 + xqy + x = 0, since xq2+q+1 = 1. Hence for every x ∈ Pq there are exactly q + 1 elements y ∈ Pq such that Fq(x, y) = 0. Now let x1, x2 ∈ Pq such that Fq(x1, y) = Fq(x2, y) = 0 and x1 6= x2. Then q+1 q+1 (x1 − x2 )y − (x1 − x2) = Fq(x1, y) − Fq(x2, y) = 0 2 The order of x1/x2 divides q + q + 1 and is not one. Furthermore q + 1 and 2 q+1 q+1 q+1 q + q + 1 are relatively prime.
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