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TRANSFORMATIONS OF CODES AND GEOMETRY RELATED TO VERONESEANS

David G. Glynn Abstract. There is a chain of polynomial codes that contains the simplex code of the projective plane over GF (q). It is related to Veroneseans of the plane. We show how to construct information sets for some of these codes using any dual hyperoval in such a plane. Also, the more general Veroneseans of hypersurfaces of degree i of projective space are considered and, related to this, a general transformation of codes and of sets of points in projective geometry that generalizes coding theoretic duality. We call it “duality of order i”. If we ensure that a set of points is taken to another set of points in the same space then the transformation is invertible for generic sets of points. First order duality corresponds to the usual duality of codes and of matroids. Quadratic duality takes any 93 configurations to another 93 e.g. Pappus. One of the Steiner triple systems having 13 points is taken to the projective plane P G(2, 3) of order 3 using an abstract version of the third order duality. This leads to a construction of the triple system using 26 conics in P G(2, 3).

1. Introduction See [9, Chapter 1] for the following basic coding theory ideas. Definition 1.1. A linear code C with parameters [n, k, d] is the set of all vectors (codewords) of length n over GF (q) which are the linear combinations of the rows of a k × n “generator” matrix G over GF (q) of rank k. The weight of any codeword is just the number of its non-zero entries; in other words, the size of its support. The distance between any pair of codewords is the weight of the difference between these words. The parameter d is the minimum (non-zero) distance between distinct words in the code. Since the code is linear this is the same as the minimum weight of a non-zero word in the code. Definition 1.2. The linear code C has a dual code C ⊥ , which has an (n − k) × n generator matrix of rank n − k, often called parity check matrix, that is orthogonal to any generator matrix G of C with respect to the standard inner product. Thus C ⊥ is the set of all words of the same length n that are orthogonal to all words in the first code. If the first code has parameters [n, k, d], then the second has parameters [n, n − k, d0 ], and a parity check matrix of C is a generator matrix of C ⊥ and vice-versa. 1991 Mathematics Subject Classification. 05E20 14M99 51E20 51M35 94B60 05B07 05B25 05B35. Key words and phrases. code, projective plane, hyperoval, functions, duality, Veronesean, information set. Typeset by AMS-TEX

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Construction 1.1. A set S of n points generating P G(k − 1, q) corresponds to a linear code with parameters [n, k, d]. One just orders the points of S in some way, and constructs a generator matrix for the code, so that each column corresponds to the homogeneous coordinates of a point of S. Remark 1.1. In Construction 1.1 a change of standard basis in the P G(k − 1, q) doesn’t change the resulting code. More precisely, the correspondence is between equivalence classes of ordered sets of points in P G(k − 1, q) up to homographies, and equivalence classes of linear [n, k] codes up to multiplication of the n positions by non-zero elements of GF (q). Also, for a linear code over GF (q) any non-zero word has q−1 non-zero multiples, and, together with the zero vector this collection (a subspace of rank 1 of the code) corresponds to the hyperplane of P G(k − 1, q) that has as its dual coordinates the coefficients of the linear combination of the rows of the generator matrix that gives the word. Thus the hyperplane intersects the set S of n points of P G(k − 1, q) in t points if and only if the non-zero words of the corresponding rank 1 subspace have weight n − t. Hence we have: Remark 1.2. The distance d of the [n, k, d] code coming from a set S of n points of P G(k − 1, q) is n minus the maximum number of points of S in a hyperplane of the space. Convention for this Section. Let Sr = P G(k − 1, q), the projective geometry of dimension r = k − 1 over the finite field GF (q), although many results will be valid for projective space over any field. From now on in this section, when we mention dual it is of the usual geometrical and not the coding theoretical kind of duality. Thus it corresponds to an inversion of the lattice of subspaces of the given Sr , where a subspace of dimension d has dual dimension r − d − 1. In later sections the duality will refer to the coding or matroid theoretic one: it is the latter kind of duality that we generalize in this paper. Many details about Veroneseans, i-uple embeddings, Veronese embeddings, etc. can be found in [3,5,13,14]. Here are a few basic ideas. Note that we are using some notation of Segre [11,12]. Later we shall discuss dualities coming from these varieties, although they are of a different nature to those of Beniamino Segre’s papers. Let us use Segre’s basic definitions, which are valid for any projective geometry Sr of dimension r over a field. Later we can specialize things for use in coding theory. Only a few changes are made to Segre’s notation. Note that a form is a homogeneous polynomial in many variables over a field. Its degree or order is the sum of the powers in each of its individual monomials. [11]. The totality of forms1 f of Sr of order i (r ≥ 1, i ≥ 2) is a linear system of dimension ρ = ri+1 − 1, where we let  rh =

 r+h−1 (r + h − 1)! , = r!(h − 1)! r

and may therefore be represented linearly with the points of an Sρ . The systems obtained in Sr by associating the f with given conditions have, for images in Sρ , varieties whose properties reflect those of the considered systems. 1 These

are homogenous polynomials in r + 1 variables with coefficients in the base field of Sr .

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For example, taking in Sr a point P or a hyperplane p, and fixing a general h from the numbers 1, 2, . . . , i, the totality of the f of Sr passing through P with multiplicity ≥ h or containing p as a component of multiplicity ≥ i − h + 1 are represented in Sρ by two linear spaces, that we shall denote by πh and χh respectively, having dual dimensions ρ − rh and rh − 1. When P and p vary in Sr , the spaces πh and χh describe in Sρ two algebraic systems ∞r 2 , which we call Φh and Ψh ; we shall indicate with these same letters the algebraic varieties which are the places of the points of such spaces, whose respective dimensions are ρ + r − rh and r + rh − 1, with the only exceptions of Φ1 for r ≥ 1, i ≥ 2, and of Ψi for r = 1, i ≥ 2 that are two systems ∞r of hyperplanes and that as a consequence, as places of points, comprise the entire Sρ . The singularities of the variety Φ2 have been already studied by Hilbert for r = 1 and arbitrary i [4], and I have already had the occasion to consider the Φ, Ψ and their multiple varieties for arbitrary r and i = 2, which are the Veronese varieties of index 2 [10]. Definition 1.3. The Veronesean (or Veronese variety) of index i of Sr ,
where r = k − 1, is a variety of dimension r of Sρ , where ρ = ri+1 − 1 = k+i−1 − 1, that i represents all the hypersurfaces (or forms) of order i in that space via the Veronese mapping vi : (x, y, z, . . . ) 7→ (. . . , xa y b z c . . . , ...), (a ≥ 0, b ≥ 0, c ≥ 0, . . . , a+b+c+· · · = i),
where all ri+1 = k+i−1 monomials of degree i in the k variables x, y, z, . . . appear i in the vector. Note that, by avoiding multinomial coefficients, this definition is good for fields of any characteristic, in contrast to Segre’s, which is only valid for characteristic 0, or when the characteristic is larger than i. Let us call this Veronesean variety V i (k − 1) = V i (k − 1, q), when the field is GF (q). For example, V 1 (k − 1, q) = P G(k − 1, q), while V i (1, q) is a normal rational curve (of order i) in P G(i, q). Putting h = 1 in the above, we see that the condition that a point P lies with multiplicity h = 1 on a general hypersurface of order i is the same as the hyperplane of Sρ that represents P containing the general point of the Sρ . The hyperplane (condition subspace) has dual dimension ρ − 1 since r1 = 1. Other conditions determine different collections of subspaces of various dimensions. Important Convention for this Paper. We prefer to dualize the Veronese space Sρ so that instead of hyperplanes of Sρ representing points of Sr we have points of Sρ representing points. Then the collection of points of Sr is represented by V i (k − 1), and the hypersurfaces of Sr of order i correspond to general hyperplanes of Sρ . For example, consider curves of order i in the plane S2 (k = 3), the main case that we analyse later in this paper. For conics (curves of order 2), V 2 (2, q) is the Veronese surface in P G(5, q), having q 2 + q + 1 points. In general, the Veronesean V i (2, q)
of i+2 index i of the plane over GF (q) is a variety of dimension 2 (i.e. surface) in i − 1 dimensional projective space, and it has order i2 . The last fact follows because the degenerate hypersurfaces (i.e. curves in this case) of order i corresponding to two sets of i lines in the plane (in generic position) will intersect in i2 points, and so i2 is the general number of points in which two hyperplanes and the Veronese
surface intersect. Note that a hyperplane of the Veronesean space P G( i+2 − 1, q) i corresponds to a curve of degree i of the plane. The points on the curve correspond to the points on the hyperplane that are also in the V i (2, q). 2 i.e.

of dimension r

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2. Function Codes from Simplex Codes Given any code over GF (q) it is possible to construct chains of codes of the same length but of larger rank, by considering spaces of functions on the codes. In geometrical terms it corresponds to certain algebraic transformations mapping sets of points in P G(k − 1, q) to sets of points in the Veronese variety V i (k − 1, q) of index i, i.e. corresponding to hypersurfaces of order i in Sk−1 . In general, an [n, k] code will be mapped to an [n, k 0 ] code where   k+i−1 0 k = . i This value can be a little smaller in degenerate cases, which occur when k is fixed and i is large. Many results in this section can also be found in the paper [15], and so this section comprises a brief survey of some of these results. We investigate k = 3. The corresponding geometry is the plane over the field GF (q). The most basic code to begin with is the simplex code S with parameters [q 2 + q + 1, 3, q 2 ] over GF (q): the columns of its generator matrix come from listing each point of P G(2, q). The dual code S ⊥ is the Hamming [q 2 + q + 1, q 2 + q − 2, 3] code. Definition 2.1. Let S 0 be the [n, 1, n] linear code over GF (q) generated by the word of all 1’s, where n = q 2 + q + 1. We define S 1 := S, the simplex code over GF (q). S is the space of linear functions on the points of P G(2, q). Next define S 2 as the space of (homogeneous) quadratic functions on the points. Then S 3 is the space of cubics on the points of P G(2, q), and we can continue for any i to define S i . Notice that these functions are defined only when we fix the representative coordinates of the points, not multiplying them by a constant. But, changing the representative coordinates will give equivalent codes.3 Note also that the set of points in space corresponding to the code S i is the Veronese variety V i (2, q) of index i. Here we summarize the coding-theoretic parameters of these Veronesean codes for the plane. Note 2.1. S 1 is a [q 2 + q + 1, 3, q 2 ] code. Since the conic with the largest number of points is the line-pair, having 2q + 1 points, it follows that: Note 2.2. S 2 is a [q 2 + q + 1, 6, q 2 − q] code, The cubic with the largest number of points is the set of 3 distinct concurrent lines (passing through a common point), and it has 3q + 1 points. Thus Note 2.3. S 3 is a [q 2 + q + 1, 10, q 2 − 2q] code if q is large enough. But things don’t go to plan if q is small, e.g. q = 2, for the dimension of the code cannot be greater than its length. We shall now state some more results about these kinds of codes. Result 2.1. If 0 ≤ i ≤ 2q − 2, (i 6= q − 1), then (S i )⊥ = S 2q−2−i . Thus, in terms of matroids, the dual of V i (2, q) is V 2q−2−i (2, q). Result 2.2. If i = q − 1, then (S q−1 )⊥ is generated by S q−1 adjoined by the word of all 1’s, S 0 . 3 However,

matter.

if the functions had degree q − 1 the actual coordinate representation wouldn’t

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Result 2.3. If 0 ≤ i ≤ q, then S i is a [q 2 + q + 1, k, q 2 − q(i − 1)] code with  k=

 i+2 . 2

Result 2.4. If q + 1 ≤ i ≤ 2q − 2, then S i is a [q 2 + q + 1, k, 1] code with  2q − i k =q +q+1− 2 2



There are some interesting special cases: i = q − 2 (dual to the case i = q) and i = q − 3 (dual to i = q + 1) that will be looked at briefly in the next section. It is not clear if the minimal weight words have been classified for all these codes. However, that project is beyond the scope of the present paper. 3. Dual Hyperovals and Bases of Codes Assume that q = 2h , h ≥ 1. A hyperoval of P G(2, q) is a set of q + 2 points, no three collinear. It corresponds to an MDS code4 with parameters [q + 2, 3, q], meeting the Singleton bound, d ≤ n − k + 1. There are various constructions of infinite sequences of hyperovals (with increasing q), the classical one being the conic plus nucleus (or “nonic”). However, they are still not classified in general, except for small cases up to about q = 32 (by computer). See [1]. An even code over GF (q), q even, is a code in which every word has even weight. The following is easy to see from the geometry: The set of columns of a [q + 2, 3, d] code over GF(q) is a hyperoval if and only if the code is even. Let H be a dual hyperoval5 of P G(2, q). This is a set of q + 2 lines
such that no q+2 three are concurrent. The number of points on the lines of H is 2 . These could be called the exterior points of H. An information set for a linear [n, k] code corresponds to a set of k independent columns (i.e. basis) of the generator matrix. In the case of the code S i an information set corresponds to a subset of the points of V i (2, q), such that specifying the values of a function of degree i at the corresponding points of P G(2, q), determines the entire function on the plane. Theorem 3.1. The exterior points of any dual hyperoval H form an information set for the code S q . Proof. Multiplying any q lines of H together we obtain a function of degree q whose value is zero at all external
points except one. Thus we obtain a basis for the code q+2 consisting of these 2 functions, and we do know that this number is the rank of the code, S q , by Result 2.3.  Since a basis for a dual matroid, or dual code, is the complement of the basis for the matroid, or code it follows that: 4 “maximum 5 This

distance separable” duality is the usual projective geometry one, not the coding or matroid theory one.

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Corollary 3.1. The interior points (not on any line) of H form an information set for the code S q−2 . Remark 3.1. There are precisely 3 independent functions of degree q+1 that are zero on all points of the plane. These can be taken as the products of q + 1 lines through three independent points of the plane, or as fxy := xy q − yxq , fxz := xz q − zxq , and fyz := yz q − zy q . We can write the sums of these as (x, y, z)A(xq , y q , z q )t , where A is any 3 × 3 skew-symmetric matrix over GF (q), with zeros on the main diagonal. Note that for points over GF(q), xy q reduces to xy and so on. This is useful to specify the code S q+1 , which is the set of all functions of degree
q + 1, modulo the ideal hfxy , fxz , fyz i. Thus this code has rank q+3 − 3, which is 2
q−1 q−3 consistent with it being the dual of the code S having rank 2 , since 

   q+3 q−1 −3+ = q 2 + q + 1. 2 2 4. Higher Order Duality

Consider any algebraic variety (or set of points) in a projective space. We say that x points “determine” the variety (or class of varieties) if, given any x points in the space, there is at least one linear collineation (homography) of the space that takes the variety to a homographic image containing those points. Further, there is at least one set of x + 1 points for which there is no homographic image containing those points. Thus x is maximal with respect to this property. For example, “2 points determine a line”, “3 points determine a plane”, “5 points determine a conic”, “6 points determine a twisted cubic in 3-d space”, “k + 3
i+k−1 points determine a normal rational curve of k-dim space”, and “ i − 1 points determine a general hypersurface (form) of order i of Sk−1 ”. Theorem 4.1. The number of points that determine the Veronesean V i (k − 1) is
i+k−1 + k. i Proof. 6 The number of parameters of the group of the space in which the Verone
2 sean is embedded is i+k−1 − 1, while the number of parameters of the group of i the Veronesean itself is the same as that of the projective geometry of dimension k − 1, which is k 2 − 1. To say that a variety of dimension k − 1 lies on a specified point of n-dimensional space is to restrict that variety by n − k + 1 parameters. Hence, the number of points needed to uniquely determine the Veronese variety of index i corresponding to Sk−1 is i+k−1 2 i
i+k−1 i




− k2 −k

 =

 i+k−1 + k. i

 Now we can describe the i’th order dual transformation: first take a set of n points in P G(k − 1, q). Use the i’th order Veronese transform vi to obtain a set 6 It

is possible that this argument breaks down for certain fields, but it will be valid if a field extension is used.

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of n points in the higher space. Write these as the columns of a generator matrix of a linear code. Then find the dual code, which is given by an m × n generator matrix that is orthogonal to the first. It turns out that this transformation will end up in the same
space as the first (dimension k) if the number of chosen points n is equal to i+k−1 + k. But this is precisely the number of points that determine the i Veronesean in the higher space. So this means that the mapping should be 1-1 for almost all sets of n points in P G(k − 1, q).7 Assumption. From now on we assume that  n=

 i+k−1 +k i

so that i’th order duality, that is the composition of the i’th order Veronese map and the matroid dual, transforms a set of n points generating Sk−1 into a set of n points in the same space. We call this transformation di . Definition 4.1. The transformation d−1 is the inverse
of the i’th order dual transi formation, di . To achieve this consider a set of i+k−1 + k points of P G(k − 1, q). d
i+k−1 Construct the (matroid) dual set of points in − 1 dimensional space for i example by finding the dual code (or equivalently the dual matroid). Then find the (usually) unique Veronese variety passing through these points in the higher dimensional space. Now there is a unique way to go back to k-dimensional space once we have the points in the Veronese variety. Proviso. It turns out
that the mapping di is not always 1-1. This happens because some sets of i+k−1 + k points in the higher space are not contained in a unique i Veronese variety. Examples are in the next section. Note that di preserves the ordering of the points of the mapped set. This is because it is the composition of the i’th order Veronese mapping and the matroid dual mapping. That is why we can now discuss the image of the “complement” of a set of points under the order i mapping.
Theorem 4.1. Let X be a set of points of P G(k − 1, q) of size i+k−1 + k, and i consider a minimal subset Y of points of X, such that, for each point P ∈ Y , every hypersurface of order i that passes through Y \ {P } also passes through P . We say that this is a circuit of order i. Then the image of the complement X \ Y under di is the set of points on a hyperplane in the i’th order dual configuration, and conversely. Proof. Under the i’th order Veronese map Y corresponds to a set of points on the Veronesean that is a first order circuit of that space. This is because hypersurfaces of order i of Sk−1 correspond to hyperplanes of the Veronesean space. In matroid theory a circuit is taken to the complement of a hyperplane in the dual, and conversely. Thus, when we go from the Veronesean space back to Sk−1 using matroid duality, the image of Y is the complement of a hyperplane.  7 Strictly

in Sk−1 .

speaking the mappings are of homographic equivalence classes of subsets of n points

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5. Examples Example 5.1: i = 1. Because v1 is the identity map, the first order dual transform is just the usual duality (of coding theory or matroids). Since we are assuming here that a set of points maps back to the original space, it takes a [2k, k] code to its dual, another [2k, k] code, and the two are not usually the same: equivalently, the corresponding sets of 2k points are usually not homographically equivalent. Example 5.2: k = 2 and i = 2.

3 In this case the number of points n = i+k−1 + k = i 2 + 2 = 5. A set of 5 points on the projective line is taken to another set of 5 points. First we use the second order Veronese map to get 5 points on the conic in the plane. Then we find the dual of these points, so getting back to five points on the line again. It can be checked that the image of the (ordered) set of five points is isomorphic to the input set. Thus the second order dual transformation of five points on the line is the identity map. Possibly, any i’th order dual transformation of subsets of i + 3 points of the line to itself is a linear isomorphism (perhaps of order 2 or the identity). We leave this as a problem for the reader. Example 5.3: k =
3 and i =
2. For the second order dual map of the plane we must have n = i+k−1 + k = 42 + 3 = 9 points. Thus a configuration of 9 points in i the plane is taken by the second order dual transform to another configuration of 9 points in the plane. Very interesting things happen! Consider, for example, the images of the three 93 configurations.8 One can show that one of the non-Pappus 93 ’s is taken to Pappus, while the others are preserved by the mapping. Let us look at these cases in more detail. We recall that a 93 configuration is an incidence structure having 9 points and 9 lines of three points each, with three lines through each point. There are embeddings in planes over most fields. The Pappus 93 is a “theorem”, in that the last incidence is automatic, once the first 9.3 − 1 = 26 incidences between points and lines are satisfied. The second 93 configuration has a regular group Z9 acting transitively on both the points and lines, and hence it may be constructed combinatorially as a difference set D = {1, 4, 6} in Z9 . The points are the elements of Z9 , while the lines are the subsets i + D, where i ∈ Z9 . The third 93 does not have a transitive group. Here, we write the points as {1, . . . , 9}, and the blocks are the subsets 268, 379, 148, 259, 346, 157, 169, 247, 358. Using Theorem 4.1 and assuming that a certain 93 is embedded in a plane, we must search for second order circuits in the 93 . It is easy to see that the smallest can be of size 6, and then they are made up of two disjoint lines of size 3. For any conic passing through 5 of the points, being degenerate, must contain both complete lines and so all 6 points. There can be no larger circuits, since the number of independent points of second order can be at most 5, while a second order circuit of size 4 must be a set of 4 collinear points, which doesn’t appear in the 93 , assumed to be in general position. Even smaller circuits of second order also cannot appear, since the 9 points are distinct. It is possible that other second order circuits of size 6 could appear: being six distinct points on a non-degenerate conic. However, these are not linear conditions and such a conic is not included in the definition of a 93 . 8 See

[2] for the latest results and references about 93 configurations, the more general nk configurations, and embedding theorems.

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Thus, we should classify the pairs of skew lines in the 93 . It turns out that for any line of any 93 , there are two other lines that are skew to it. Thus one can construct chains of lines, each skew to the next, coming back to the start. Every line of the 93 will be in a chain. To work out the combinatorial structure of the second order dual of the 93 we then take any pair of consecutive lines of a chain. In return the three points in the complement of these lines will be a line of the dual structure (another 93 ). Now the Pappus 93 has three chains, each of size 3. For any pair of (consecutive) lines in one of these chains, the complement is the third line of the chain. Thus we immediately see that the second order dual of Pappus is the same configuration. Consider the second 93 (that has a transitive automorphism group). Using the difference set construction above, there is one chain of length nine: [146, 359, 248, 137, 269, 158, 479, 368, 257], which produces, as complements of consecutive lines, [278, 167, 569, 458, 347, 236, 125, 149, 389] : a new 93 design. However, we can calculate that if D := {1, 4, 6} (mod 9), then the complement of the consecutive skew lines D and D + 1 is E := Z9 \ D \ (D + 1) = 7D + 2 = {3, 8, 9}. Thus the second order dual 93 is isomorphic to the second 93 by multiplying all points by 7. Since 73 ≡ 1 (mod 9) in fact the second order mapping d2 satisfies d32 = id. Next consider the third 93 above. It is found that there are two chains, one of length 6 and the other of length 3: [157, 286, 379, 148, 259, 346], and [169, 247, 358]. Consecutive lines then give the sets: [349, 145, 256, 367, 178, 289], and [247, 358, 169]. This “new” 93 is Pappus, since we can group these new “lines” into three chains of consecutive skew subsets: [349, 256, 178], [145, 367, 289], [247, 358, 169]. Thus the image of the third 93 , under the second order dual transform, is the first 93 , which is Pappus. The reason that the dual transform is not 1-1 in this case is that there is more than one Veronese surface in S5 passing through the matroid dual of the 9 points of the Pappus configuration. By varying the Veronese surface we can obtain either an image of Pappus or of the third 93 under the second order Veronese map from the plane. Example 5.4: Let k = 2 and i = 3. Consider the order 3 dual transformation of sets of 13 points of the plane. In this case an interesting kind of third order circuit

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is given by a “grid” of 9 points lying on three skew lines of three points each, and also on another set of three skew lines transversing the first three. Each set of three lines is a degenerate cubic curve. Since a general pair of cubics intersect in 9 points, while 9 points also determine at least one cubic curve, these nine points on the six lines form an “associated” set of points which respect to cubics. Thus any cubic passing through 8 of the points will necessarily pass through the ninth. A philosophically interesting structure of 13 points to consider is that of a Steiner triple system, or ST S(13). This is a so-called 2-design, having 26 subsets, each of size 3, such that any pair of points is in a unique block. There are also six blocks passing through each point. Even though we can show (below) that such an ST S(13) cannot be embedded (with distinct points and lines) into a projective plane over a field, we can neverthe-less contemplate an abstract third order duality acting on it. This will produce a structure having 13 points, and a number of 4-point lines, which have to be calculated by considering the complements of the third order circuits of the ST S: the subsets of nine points on two distinct pairs of three skew lines as above. Call this a regulus of the ST S.9 There are actually two non-isomorphic ST S(13), first constructed by Zulauf [16]. The first ST S(13), denoted by ST SA here, has a point-transitive, but not blocktransitive10 , automorphism group of order 39. We can construct it by using two partial difference sets {7, 10, 11}, {4, 6, 12}, and cycling in Z13 . Using magma [8], it is quite easy to classify all the reguli. The way we used was simply to look at all 3-subsets of the 26 lines, and to sort out which ones covered the same sets of 9 points. The result was that there are 13 reguli in ST SA with two sets of skew lines: all reguli are shifts mod 13 of the following representative regulus: {0, 2, 8}, {3, 5, 11}, {4, 6, 12}, and {0, 3, 4}, {2, 5, 6}, {8, 11, 12}. These cover the point-set: {0, 2, 3, 4, 5, 6, 8, 11, 12}. There are also two further lines contained in the same set of points, which are: {0, 6, 11}, and {2, 3, 12}. It is to be noted that the configuration of nine points and eight 3-point lines induced by this subset is precisely a Pappus 93 , minus one of its lines; see Example 5.3 above. (The ninth line would have to be {4, 5, 8}, but this is not a line of ST SA.) Since Pappus is a theorem for a plane over a field, this shows that the ST SA cannot be embedded in a nice way (using distinct points and lines) into any plane over a field. However, as indicated before, we can never-the-less continue with an abstract version of the third order dual, and obtain a structure of 13 points and 13 “lines” (perhaps embedded in a plane), where the lines are the complements of the 9 points covered by the reguli. Since Z13 \{0, 2, 3, 4, 5, 6, 8, 11, 12} = {1, 7, 9, 10}, we see that G := {1, 7, 9, 10} is a difference set for the structure. It turns out that this is the projective plane of order 3, because {g − g 0 | g, g 0 ∈ G, g 6= g 0 } = Z13 \ {0}. The inverse mapping is immediately obvious. It is related to the problem of how to construct ST SA from the projective plane of order 3. Let us give an answer 9 The

plural of regulus is “reguli”. are two block orbits, each of size 13.

10 There

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in concise form. Let G be a difference set for P G(2, 3), mod 13. Then one can calculate that 2G and −2G are two sets of 4 points in the plane, that are nondegenerate conics (of 4-arcs). With respect to each conic, there is a set of 3 interior points (each point lying on two chords of the conic). These two sets of three points will be the partial difference sets for an ST SA, and the image of that ST SA under the third order dual will be the projective plane order order 3 with difference set G. Since there are many cyclic (but equivalent) cyclic Singer groups of order 13 acting on the P G(2, 3), we see that the inverse transform d−1 3 back to the ST SA is not uniquely defined. The group of ST SA is of order 39, and the above construction from a difference set of P G(2, 3) shows why. A planar difference set has a “multiplier” group11 . For a plane of order q any divisor m of q gives such a multiplier, and it is possible to assume that the difference set is closed under multiplication by m. In our case we have a plane of order 3 and so m = 3 is a multiplier for the difference set. Since 33 ≡ 1 (mod 13) this multiplier gives another automorphism of order 3 for the difference set, and we obtain the (non-abelian) group of order 39 = 13.3 of P G(2, 3), that induces the full automorphism group of the Steiner triple system ST SA on 13 points. The analysis of the second ST S(13), ST SB, proceeds similarly. But since the group of this structure is only of order 6 (being isomorphic to the symmetric group on three letters), the details are more complicated. It turns out that there are also 13 reguli in the ST SB, and this also leads to a structure of 13 points and 13 lines, each having four points each. Further, there are four lines through each point in this structure. One might surmise that this another P G(2, 3), but this is false, since these “lines” may intersect in 0, 1, or 2 points. We leave it as an exercise to investigate the reasons why ST SB may have the same number of reguli as ST SA. Example 5.5: Let k = 4 and i = 2. Then we get a transformation d2 of sets of 14 points of 3-dim space. It is left to the reader to examine this case, and to find other interesting applications. References 1. Hyperoval Website: www.math.cudenver.edu/∼wcherowi/research/hyperoval/hytoc.htm. 2. D.G. Glynn, A note on nk configurations and theorems in projective space (submitted to Bull. Austral. Math. Soc., preprint at http://internal.maths.adelaide.edu.au/people/rcasse/). 3. R. Hartshorne, Algebraic Geometry, Springer, 1983. 4. D. Hilbert, Ueber die Singularit¨ aten der Diskriminantenfl¨ ache, Math. Ann 30 (1887), 437– 441. 5. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Second Edition, Oxford University Press, Oxford, 1998. 6. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991. 7. Peter Kaski, A Census of Steiner Triple Systems and Some Related Combinatorial Objects, Helsinki Univ. Techn. for Theoretical Computer Science, Helsinki, 2003 (Research Reports 78). 8. Magma, Computer Algebra System, Univ. Sydney. 9. V.S. Pless, W.C. Hoffman and R.A. Brualdi, Handbook of Coding Theory, Elsevier, 1998. 10. B. Segre, Sulle variet` a di Veronese a due indici, Rend. R. Acc. Naz. Lincei (6) 23 (1936), 303–309, 391–397. 11 first

discovered by Marshall Hall Jr.

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DAVID G. GLYNN

11. B. Segre, Un’estensione delle variet` a di Veronese, ed un principio di dualit` a per forme algebriche. Nota I, Rendiconti dell’Academia Nazionale dei Lincei (8) 1 (1946), 313–318. 12. B. Segre, Un’estensione delle variet` a di Veronese, ed un principio di dualit` a per forme algebriche. Nota II, Rendiconti dell’Academia Nazionale dei Lincei (8) 1 (1946), 559–563. 13. J.G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford, 1985. 14. I.R. Shafarevich, Basic Algebraic Geometry, Springer, 1994. 15. Anders Bjaert Sorensen, Projective Reed-Muller Codes, IEEE Transactions Inf. Th. 37 (1991), 1567–1576. ¨ 16. K. Zulauf, Uber Tripelsysteme von 13 Elementen, Dissertation Giessen, Wintersche Buchdruckerei, Darmstadt, 1897. School of Mathematical Sciences, University of Adelaide, SA 5005, Australia E-mail address: [email protected], [email protected]