Using ideas from algebraic coding theory, a general notion of a derivation set for a projective

Transcription

1 TRANSLATION PLANES AND DERIVATION SETS* Dedicated to Professor Tallini on the occasion of his 60 th birthday E. F. Assmus, Jr. and J. D. Key Using ideas from algebraic coding theory, a general notion of a derivation set for a projective plane is introduced. Certain geometric codes are used to locate such sets. These codes also lead to upper bounds for the p-ranks of incidence matrices of translation planes in terms of the dimensions of the associated codes.. Introduction. The main purpose of this paper is to place the notion of a derivation set for a nite projective plane in a coding-theoretic setting. In doing so, we expand on a remark made in [] concerning \generalized derivations" for translation planes. Although the notion of a derivation set is of a general nature, applying to any projective plane, its most frequent use has been in the construction of translation planes. We choose, therefore, to link these two topics, thus naturally introducing certain geometric codes. These codes then yield an upper bound for the p-rank of an incidence matrix of a translation plane of order a power of p in terms of the dimension of the associated geometric code. In [8, Theorem 5] Ostrom dened a very general notion of derivation for projective planes; with the help of geometric or combinatorial arguments, new planes have been constructed using this notion, but most of the new planes involve the choice of a Baer segment of a line as a derivation set. The theory of projective planes proposed in [], involving codes associated with the plane and its ane parts, leads naturally to a notion of a derivation set that is less general than that proposed by Ostrom, but more general than the geometric derivations most frequently used. Our denition (see Section 4) involves describing a subset D of points of a line L of a projective plane with code C as a derivation set if there exists a set of constant vectors in C + C?, all of the same weight, all with D in the support, and any two having at most one common point o L in their supports. In fact, from our denition it follows that for a plane of order p 2, p a prime, the only possible non-trivial derivation sets are the Baer segments and that for planes of prime order, there are no non-trivial derivation sets at all. *This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. Typeset by AMS-TEX

2 2 Concerning bounds, we show in [] that the incidence matrix of any ane translation plane of order q = p s, where p is a prime, has rank over F p bounded above by q + q 2? dim(b(f q jf p )) where B(F q jf p ) is the code over F p of the design of points and s-ats of the ane geometry AG 2s (F p ). We can write this upper bound as q + dim(b(f q jf p )? ). We show here (see Section 3, Theorem 2) that B(F q jf p )? is the code spanned by all vectors of the form v X? v Y, where X and Y are parallel s-ats and v Z denotes the characteristic function of a subset Z of points. Denoting this latter code by E(F q jf p ), this bound has a renement for translation planes with \kern" (see [6]) a subeld F of F q, namely q + dim(e(f q jf )); where here the generating vectors, v X? v Y, must have X and Y parallel s-ats coming from a subspace of V viewed as a vector space over F, rather than F p : see Section 3, Theorem. Thus, for example, the bound for q = 6 is 0, rather than 09, when the translation plane is \2-dimensional". All translation planes of order 6 are known (see [8]), and all the 2-ranks have been determined (see [7], or Section 3). The (ane) ranks lie between 8 (the desarguesian) and 05 (the Lorimer and the derived semi-eld planes), with the 2-dimensional planes having rank 97. We do not, unfortunately, have any instances of planes meeting these new bounds, but if there were any such planes, they would all be linearly equivalent (see Section 2), just as in the case of dimension equal to q + dim(b(f q jf p )? ). The formulae for the dimension of the spaces B(F q jf ) and E(F q jf ) given by Delsarte [6] and Hamada [9] (see also [, Appendix I]) are dicult to work with. We quote a formula, due to Key and Mackenzie [3], for dim(b(f q jf p )) that is easy to use, but we do not have an easy-to-use formula for dim(e(f q jf )) in the general case. 2. Preliminaries. We will adhere to the notation of [] which we now briey review. Let be an arbitrary projective plane of order n, p a prime dividing n. Set N = n 2 + n + and view F N p as the vector space of all functions from the point set of to the eld F p. Let C p () be the subspace of F N p generated by the characteristic functions of the lines of, C p ()? the orthogonal to C p () under the standard inner product, and set B p () = C p ()+C p ()?. It is shown in [] that B p () has minimum weight n + and that the minimal-weight vectors are precisely the scalar multiples of the characteristic functions of the lines of. If is an ane plane of order n and p a prime dividing n, we make analogous denitions: B p () = C p () + C p ()? where C p () is that subspace of F n2 p generated by the characteristic functions of the lines of. From [], B p () has minimum weight n, and its minimal-weight vectors are scalar multiples of the characteristic functions of certain subsets of points of. Now, however, subsets of points that yield minimal-weight vectors are not necessarily, in fact not usually, lines. If is the projective completion of with L

3 the line at innity, we will write = L, and then B p () is the image of B p () under the projection map that ignores the points of L. In the ane case, B p ()?, called the hull of and denoted by Hull p (), plays a signicant role. We have Hull p () = C p () \ C p ()?, and it is shown in [] that Hull p () is generated by vectors of the form v l?v m where l and m are parallel lines and v X denotes the vector in F n2 p that is the characteristic function of X. Here again, of course, we are viewing F n2 p as the vector space of all functions from the points of to the eld F p. From [], dim(hull p ()) = dim(c p ())? n. Next let q = p s, set K = F q, and view V = K K as a 2s-dimensional vector space over F p. Then jv j = q 2, and F V p, the vector space of all functions from V to F p, is a q 2 -dimensional vector space over F p. Now, for any subeld F of K one can view V as a vector space over F. An s-dimensional at of V (i.e. a coset of an s-dimensional subspace of V viewed as a vector space over F p ) that is a coset of an F -subspace of V will, for convenience, be called an F -at. Let B(KjF ) be that subspace of F V p generated by all v X, X an F -at. It is shown in [6] that B(KjF ) has minimum weight q and that its minimal-weight vectors are precisely the scalar multiples of its generating vectors v X. We have a Galois correspondence between the subelds of K and the codes B(KjF ). Here, B(KjK) is the smallest code and can be viewed as C p (AG2(F q )), where AG2(F q ) denotes the desarguesian ane plane of order q = p s. In [] we show that an ane plane of order q = p s is a translation plane if and only if C p () is isomorphic to a subcode C of B(KjF p ) for which, viewing C as C p (), Hull p () C p () B(KjF p ) B p (): It can happen, of course, that C could be taken to be in B(KjF ) for a subeld F of K (in the literature on translation planes one would say that had \kern" F ); when this happens we say that is contained in the code B(KjF ). Then we have Hull p () C p () B(KjF ) B(KjF p ) B p (): The nal notion we need from [] is that of linear equivalence: if and 2 are two ane planes of order n and p is a prime dividing n, then and 2 are linearly equivalent (at p) if Hull p () and Hull p (2) are code-isomorphic. Notation new to the present paper: E(KjF ) is that subspace of Fp q2 generated by the vectors of the form v X? v Y where X and Y are parallel F -ats of V. Clearly E(KjF ) B(KjF ). Note: we will omit the subscript \p" that appears, for example, in C p (); Hull p () and B p (), writing simply C(); Hull() and B(), where p is understood to be a xed prime dividing the order n of the plane. For translation planes, certainly, n is a power of p Bounds. In [] we proved that any translation plane of order q = p s satises dim(c()) q + q 2? dim(b(f q jf p )):

4 4 Put another way, the p-rank of an incidence matrix of is bounded above by q + q 2? dim(b(f q jf p )). Moreover, any two translation planes meeting this bound are linearly equivalent. We prove here the following analogous result. Theorem. Suppose is a translation plane contained in B(F q jf ), where F is a subeld of F q. Then dim(c()) q + dim(e(f q jf )): Moreover, any two translation planes contained in B(F q jf ) and meeting this bound are linearly equivalent. Proof. If C() is isomorphic to C B(F q jf ), then Hull() is isomorphic to a subcode H of E(F q jf ); this follows since Hull() is generated by vectors of the form v l? v m, where l and m are parallel lines of, and, under the isomorphism of C() onto C, v l will correspond to a vector v X that is a generator of B(F q jf ), these being the only (up to scalar multiples) weight-q vectors of B(F q jf ). Since dim(h) = dim(hull()) = dim(c())? q, the required inequality follows. Moreover, equality implies that H = E(F q jf ), and hence any two ane translation planes contained in B(F q jf ) and meeting the bound have hulls isomorphic to E(F q jf ), and hence are linearly equivalent. In order to show that the bound is, in fact, the same as that of [] for F = F p, we rst determine B(F q jf p )?. The following result is implicit in the work of Delsarte [6], but, for the convenience of the reader, we include a proof. Note that a neat proof, via the modular algebra approach to Reed-Muller codes, can be found in [5, Chapter 3]. Theorem 2. B(F q jf p )? = E(F q jf p ). Proof. First of all, it is clear that the generators of E(F q jf ), for any subeld F of F q, are in B(F q jf p )?, since an s-at of V meets any two parallel s-ats in the same number of points modulo p ; i. e. in one point of each at of the parallel class when the corresponding subspaces have intersection f0g, and in the empty set, or a power of p points, otherwise. Thus E(F q jf p ) B(F q jf p )?. To prove equality we will prove that the two dimensions are equal. We use the radix-p form of the dimensions of E(F q jf p ) = E and B(F q jf p ) = B as given by Delsarte [6]. There it is shown that, for q = p s, the dimension of B is equal to the number of integers z with z < p 2s that \contain", in the sense of Delsarte, s multiples of (p? ), and that the dimension of E is equal to the number of integers z with z < p 2s that \properly contain" s multiples of (p? ). (In Delsarte's notation B is C p? (; ; : : : ; ) with s ones and E is C(; ; : : : ; ) with s + ones.) More P generally, for the moment, let z be a positive integer with p-ary expansion z = i=0 a ip i where, of course, 0 a i < p and almost all a i are 0. For z to contain (p? ) k times we must be able to write a i = a () i + + a (k) i with 0 a (j) i < p for all P i and j in such a manner that there are positive integers m; : : : ; m k with m j (p? ) = i=0 b(j) i p i

5 and P b (j) i a (j) for all i and j. Now, by a \casting out nines" argument, an integer x = i=0 x ip i is divisible P P by (p? ) if and only if i=0 x i is. P Suppose that z = i=0 a ip i contains (p? ) k times and that i=0 a i = k(p? ). Then, however one writes P a i = a () P P i + + a (k) i with m j (p? ) = i=0 b(j) i p i and a (j) i b (j) i, we must have i=0 a(j) i i=0 b(j) i = n j (p? ), with n j a positive integer, and hence P P P i=0 a(j) i (p? ) for all j. Since i=0 a i = k(p? ), we have thus i=0 a(j) i = (p? ) for i j k and so a (j) i = b (j) i for all i and j, where each n j =. In this case z exactly contains (p? ) k times and hence not properly. Now let B and E be, respectively, the sets of integers giving the dimensions of B and E. Then, from P P what we have shown above, and writing z = 2s? i=0 a ip i, we have z 2 B if and only if 2s? P i=0 a i s(p?), and P P z 2 E if and only if 2s? P i=0 a i > s(p?). Let E = fzjz 2 Eg where, if z = i=0 a ip i, z = i=0 (p?? a i)p i. Since 2s? i=0 (p?? a i) = 2s(p? )? P 2s? i=0 a i, it follows that B and E are disjoint with B [ E = fz 2 Z j 0 z < p 2s g. Hence dim(b) + dim(e) = p 2s = q 2 since, obviously, jej = jej. This implies dim(e) = dim(b? ) and completes the proof. Remarks. ) The Galois correspondence between subelds of F q and the E(F q jf ) or B(F q jf ) yields a hierarchy of translation planes corresponding, in the language of the existing literature on translation planes, to the \kern" or \kernel" of the translation plane. Theorem shows that this hierarchy reects itself in the p-rank of the incidence matrix of the translation plane. Roughly speaking, the rank goes down as the kernel gets larger. Of course, Theorem simply gives upper bounds that grow smaller as F gets larger; in practice the p-rank is rather sporadic: see [, x7] and the Table of Ranks below. For a full description see [7]. Table of Ranks The ranks of the ane translation planes of orders 9, 6, and q q + dim(e(f q jf q )) q + dim(e(f q jf p q)) q + dim(e(f q jf p )) C() = B(F q jf q ) C() B(F q jf p q) C() B(F q jf p ) 9 Bounds Ranks Bounds Ranks 8 97(2) 97,99,0,05(2) 25 Bounds Ranks ,250,252,254, 255,256,257(3),258(4), 259(2),260(2),26(2), 263,264 2) We do not, unfortunately, have an easy-to-use formula for dim(e(f q jf )) where F is an arbitrary subeld of F q : we have simply used the formulae given by Delsarte and Hamada.

6 6 A formula for dim(b(f q jf p )) that is very simple to use is derived in [3] from Hamada's general formula: dim(b(f q jf p Xs? )) = (?) i 2s p(s? ) + s i 2s i=0 where, as usual, q = p s. Of course, by Theorem 2, dim(e(f q jf p )) = q 2?dim(B(F q jf p )). 3) It would be useful to have a formula for dim(e(f q jf p q)) for q an even power of p. We have computed the dimension of E(F6jF4) using Delsarte's method; it is 85 and yields the bound 0 for 2-dimensional translation planes of order 6. There are precisely two such planes (see [4]); they both have 2-rank 97. It would be interesting to know whether or not they are linearly equivalent; had they both had 2-rank 0, they would have been, by Theorem. 4) Mackenzie [7] has also obtained, from Hamada's general formula, the following: dim(b(f p 4 jf p 2)) = 4 p p p6 + 8 p5? 9 p4 + 8 p p2 : 4. Derivation Sets. Let be an arbitrary projective plane of order n, and p a prime dividing n. Let L be a line of. We want to explain when a subset D of L will be called a \derivation set" for. Set = L. Then is also of order n, and we have the natural projection of B() onto B(), where again we will omit the subscript \p" in what follows. If dim(b()) = k, then B() is an [n 2 ; k? ] code of minimum weight n. Besides the characteristic functions of the lines of, there are, usually, other minimal-weight vectors in B(), all of which are of the form v X, where X is a subset of points of, jxj = n, and 2 F p? f0g. Lemma of [] describes the nature of such minimal-weight vectors of B(): if b = v X is a weight-n vector of B(), where X is not a line of, normalized so that =, then lines in the same parallel class of meet X constantly modulo p, and, if r is the number of classes whose lines meet X in 0 (mod p) points, then there is a unique vector b 2 B() with b =2 C()? such that b = v X for some subset X of points of, b projects to b, and weight(b) = jxj = n + r, with < r < n, r (mod p). The set L \ X is, obviously, of cardinality r. For example (see []), if = P G2(F q 2), X could be the points of a Baer subplane, with L \ X a Baer segment of L. Then X would be an ane Baer subplane of AG2(F q 2), furnishing a weight-q 2 vector of B(AG2(F q 2)) that does not come from a line. (Here, of course, q is a power of p, the order of being q 2.) The above discussion should motivate our denition of \derivation set". Denition. Let D be a subset of points on a line L of a nite projective plane of order n. We say D is a derivation set for if there is a prime p dividing n and a collection D of vectors of B() satisfying the following conditions:

7 (i) jdj = njdj; (ii) each vector b 2 D is of the form v X where D X and weight(b) = jxj = n + jdj; (iii) for distinct b and c in D, weight(b? c) 2n? 2. In the denition (i) simply indicates that we plan to remove jdj parallel classes of lines each class having, of course, n lines; (ii) insures that each line has n points; and (iii) insures that the \new" lines intersect properly. The point of the denition is that, using and D, it is easy to construct an ane plane, (; D), as follows: the points of (; D) are those of L ; the lines of (; D) are (i) those of coming from the lines M of with M \L =2 D and (ii) the subsets X of the points of coming from fv X jv X = im(b); b 2 Dg, where im(b) denotes the image of b under the natural projection of B() to B( L ). The proof that this denes a plane follows from Lemma of [] (stated at the beginning of this section): rst notice that the number of lines we have introduced is n 2 + n. A line m of coming from a line M of with M \ L =2 D has jm \ Xj = for each X with v X = im(b), where b 2 D. Further, condition (iii) ensures that if P and Q are points of with m the line through P and Q where m is a line of coming from M of with M \ L 2 D, then P and Q are together in a unique subset X, as introduced. It is an immediate consequence of the denition and [, Lemma ] that a derivation set has cardinality congruent to modulo p. Those with cardinality equal to have for D a parallel class of lines. Those with cardinality greater than may admit various D's (see Section 5). Notice that one could, of course, carry out the above construction using two or more disjoint derivation sets on the line L: for example, the Hall plane of order 6 can be obtained from P G2(F6) by using two disjoint Baer segments. Perhaps a more exact term for the above notion would be \primitive derivation". However, this notion does explain why Baer segments appeared so frequently as derivation sets; the next proposition, which is a direct consequence of [, Lemma 2], makes the point. Proposition. For a plane of prime order there do not exist non-trivial derivation sets. For a plane whose order is a square of a prime, the only possible non-trivial derivation sets are Baer segments. The \classical" derivation uses a Baer segment, D, of L and a collection of Baer subplanes having L as a line and D as the intersection of L with the Baer subplanes; this is, of course, one of our cases, but there are other cases as well. Our denition suggests looking for derivation sets under mild, algebraic-coding-theoretic constraints, whereas the broader denition of Ostrom [8] calls for a set-theoretic search. Consider, therefore, B(KjF ), where K = F q and F = F p. B(KjF ) is a code of length q 2 and minimum weight q, and we know all the minimal-weight vectors. If is an ane translation plane of order q, we have C() B(KjF ) B(). If we restrict the search to B(KjF ), then we would be very close to the methods of Ostrom [9]; searching in B() would, of course, be superior, but more dicult, since a survey of the weight-q vectors of B(KjF ) is more tractable than a survey of those of B(). We now show that the \classical" Baer subplane derivation, in the case of translation planes, takes place in the subspace B(KjF ) of the usually larger B(). 7

8 8 Proposition 2. If q is a square and is an ane translation plane of order q, then every ane Baer subplane of has point set X satisfying v X 2 B(F q jf p ). Proof. The result is well-known [7, 5], but we sketch the proof for the convenience of the reader. First let be an arbitrary projective plane, a Baer subplane of, and (P; L) a ag of that belongs to. Let be an elation of with centre P and axis L. Then, if Q and Q are points of where Q =2 L, it follows that is an elation of ; i.e. =. For, if A is any point of not on L and not on the line QQ, we have that AQ and P A are lines of. The point B = QA \ L is on, and hence BQ \ P A is a point of. But A = BQ \ P A, and thus A is on. We need only show now that every point C of on QQ has C on ; but this follows if we reverse the roles of A and Q, since there must be an A in not on L or QQ. Now let = L be an ane translation plane. All elations with axis L and centres on L exist. If X is the point set of a Baer subplane of and L is a line of this Baer subplane, X = X \ is an ane Baer subplane of with v X 2 B() []. We want to show that v X 2 B(F q jf p ). Without loss of generality, we can assume O 2 X, O being the zero vector of V. Suppose Q 2 X, Q 6= O. The line QO meets L at a point P (at innity) and there is an elation of with centre P that moves O to Q. By the above, this elation xes X setwise. It is dened by the vector (i.e. point) Q 0 going to Q 0 + Q and hence X is closed under addition. Thus X is a subspace of V, and v X 2 B(F q jf p ). Remarks. ) For an arbitrary ane plane of order n = m 2, the Baer subplanes will correspond to minimal-weight vectors of B p () where p is any prime dividing n []. Notice that for ane planes we use \Baer subplane" to mean a subplane coming from the projective completion and having the line at innity as a line. 2) For a translation plane of order q = p s given by the spread fs0; S; : : : ; S q g (that is, (q + ) s-dimensional subspaces of V with S i \ S j = f0g for i 6= j), a Baer subplane corresponds to a translate of an s-dimensional subspace T with dim(s i \ T ) = s or 0 for 2 all i. 3) Proposition 6 of [] and its Corollary 5 assert that a translation plane of order p 2 has at least p 3 (p 2 + )(p + ) desarguesian Baer subplanes. It is possible for it to have more: P G2(F p 2) does. The Proposition shows that if is a translation plane of order p 2 with translation line L, there are precisely p 3 (p 2 + )(p + ) Baer subplanes of that have L as a line, and all of these subplanes are desarguesian. For a 2-dimensional translation plane of order q 2 (i.e. for of order q 2 with C( L ) B(F q 2 jf q )), we know that there are at least q 3 (q 2 + )(q + ) desarguesian Baer subplanes that have L as a line, but there may be more: s-dimensional subspaces related to B(F q 2 jf p ) may provide others.

9 5. Ovals and Derivation Sets. We next turn our attention to the case p = 2. With the notation of Remark 2 directly above, it may happen that a subspace T satises dim(s i \ T ) = 0 or for all i. Such a T will have the property that v T is a weight-q vector of B(KjF ) with jt \ lj = 0;, or 2 for all lines l of the translation plane. Now if = L and b is the preimage of v T given by [, Lemma ] (and stated in section 4), then b + v L = v X where X is an oval (q + 2 points with no three collinear) of. We show how a set of ovals in a translation plane coordinatized by a neareld of even order can dene the vectors in D, yielding a derivation set D of cardinality q?. The construction depends on the existence of a hyperbolic oval in the ane translation plane. Theorem 3. Let be a projective plane of even order q. Suppose that (i) has an oval O0, and (ii) there are two points, P0 and Q0 on O0 for which is (P0; Q0)-transitive. Then, if L is the line through P0 and Q0, D = L? fp0; Q0g is a derivation set for. Proof. The condition that is (P0; Q0)-transitive that is, that every central collineation with centre P0 and axis through Q0 exists is equivalent to the condition that is a translation plane coordinatized by a neareld [7; x3..22(e)]. Thus q = 2 s, for some s. We need to produce the set D of vectors v X in the binary code B() with jxj = 2 s +?, and with X D. Since any oval O of provides a vector v O in C()? of weight 2 s + 2, the oval O0 will yield b = v L + v O 0 of weight 2 s +? with support containing D. Our goal is to extract a set of 2 s (2 s?) of these ovals, with v O +v Q of weight at least 2 s +? for distinct O and Q. This we accomplish in a purely combinatorial manner. Let H denote the group of all collineations of with centre P0 and axis passing through Q0. Then H has order 2 s (2 s? ) (see [, Chapter 6], for example) and no non-identity element of H can x O0. Thus the orbit of O0 under H has size 2 s (2 s? ). If we form the incidence structure S consisting of the points of not on L, and having for blocks the orbit of ovals, together with the lines through P0 and Q0 other than L, then we can show that this structure is an ane plane of order 2 s. Now S has 2 2s points and 2 2s + 2 s blocks, each of size 2 s. We show that any two points of S are on exactly one block of S: let P and Q be distinct points of S, not together on a line through P0 or Q0. The set of images of O0 under the subgroup of elations of H forms a parallel class of blocks, so we can assume that P is on some Q. If Q is also on Q, then we have a block through P and Q; if Q is not on Q then form the lines P Q0 and QP0 and let them intersect at R. Let S be the point of intersection of QP0 and Q. Then there is a homology in H with centre P0 and axis P Q0 that maps S to Q, and hence maps the oval Q to one through P and Q. Thus any two points of S are on at least one block, and now a count of (pairs of points, block)-intersections, in two ways, yields that any two points are together on exactly one block. The set thus gives the required set of ovals. Remarks. ) If one takes = P G2(F2 s) and for the oval O 0 a conic plus nucleus, with the nucleus either P0 or Q0, then the derivation is taking place inside B(F2 sjf 2): for the conic given by x 2 = yz, with h(0; 0; ) 0 i for the line L at innity, contains the point 9

10 0 h(0; ; 0)i = P0 of L, and its nucleus is Q0 = h(; 0; 0)i, which is again on L. In the ane plane with point set f(a; b; )ja; b 2 F2 sg, the conic consists of the points f(t; t2 ; )jt 2 F2 sg. Since we are in characteristic p = 2, this set is an s-dimensional subspace over F2. 2) It follows also (and the calculation is similar) that a conic plus nucleus will be furnished by a minimal-weight vector of B(F2 sjf 2) if and only if its nucleus is at innity. Thus the derivation, even if we restrict to ovals coming from conics, may not be taking place in B(F2 sjf 2), but in B(AG2(F2s)). Moreover, not all ovals come from conics when s > 3: see, for example, [0]. 3) Another similarity with the derivation coming from Baer subplanes is that, in the new projective plane, one has the property that any line of L that has been discarded, together with the points P0 and Q0 on the new line at innity, becomes an oval of the new plane. 4) We do not know whether or not there is a D which, for our derivation set of cardinality 2 s?, will yield a new non-desarguesian projective plane. Any D produced as in the proof of Theorem 3 will yield a translation plane coordinatized by a neareld with multiplicative group isomorphic to that of the neareld of the original plane, for the new projective plane with the new line M at innity has, in its automorphism group G, all central collineations with centre P0 and axis through Q0, and thus, by [7; x3..9, p.23], has all central collineations with centre Q0 and axis through P0. It follows easily that is a translation plane, with M a translation line: see, for example [; Theorem 4.9, p. 00]. Thus by [7; x3..22(e) and x3..34], M can be coordinatized by a neareld D, and D is unique. By [7; x3..22(b)], the multiplicative group, D, is isomorphic to the group G(P0; L ) of homologies with centre P0 and axis L through Q0, and this, by our construction, is isomorphic to the multiplicative group of the original coordinatizing neareld. Thus when the neareld is a eld, the new plane is desarguesian, however one chooses the oval. 5) In [2] Kelly gave a similar construction using even more conics; see also Bruen and Silverman [4].

11 6. Other Derivation Sets. We turn, nally, to other possible derivation sets that might arise from minimal-weight vectors of B(KjF ), where K = F q and F = F p. Suppose is a translation plane with translation line L and = L. We have, as before, C() B(KjF ) B(): If b 2 B(KjF ) has weight q but is not a line of, let b = v X be a pre-image of b with weight q + r, where r (mod p). We pointed out in [] that X is a blocking set (see [3]) for, and hence r + p q. Without loss of generality, we may assume that b = v T, where T is an s-dimensional subspace of V, and q = p s. If S0; S; : : : ; S q is the spread dening, then r = jfs i js i \ T 6= f0ggj. In general it is dicult to count the number of T with given intersection properties, and all sorts of intersections can occur: for example, a \random" look at the possibilities for a spread of one of the 2-dimensional translation planes of order 6, for example the one dened by the coordinate set M2 of [8], gave various intersection patterns for subspaces in B(F6jF2). If we let the 4-tuple [x0; x; x2; x3] correspond to a subspace T meeting x i members of the spread in subspaces of dimension i, for i = 0; ; 2; 3, so that r = x + x2 + x3, then we found instances of: [2, 0, 5, 0], r = 5 (Baer subplanes); [0, 3, 4, 0], r = 7; [8, 8, 0, ], r = 9; [8, 6, 3, 0], r = 9; [6, 9, 2, 0], r = ; [4, 2,, 0], r = 3; [2, 5, 0, 0], r = 5 (ovals). There is one case for which the count is rather easy: if S i \ T is of dimension s? for one (and hence precisely one) S i. Here we are assuming s > 2, since in the case s = 2, all T, except the S0; S; : : : ; S q, yield Baer subplanes. First observe that if dim(s i \ T ) = s? and dim(s j \ T ) 2 for some j 6= i, then, since S i \ S j = f0g, dim(t ) s + > s, which is an impossibility. It follows that dim(t \ S i ) = s? for exactly one S i, dim(t \ S i ) = for exactly p s? S i, and T \ S i = f0g for p s? (p? ) of the S i. Here r = + p s? and the b has weight p s + p s? +. Choosing such a T is easy: simply take an (s? )- dimensional subspace of one S i and a -dimensional subspace of another and let T be the (necessarily direct) sum. An easy count gives the number of such subspace to be p(p s + )(p s? ) 2 =(p? ) 2 and hence, counting translates, there are p s + (p s + )(p s? ) 2 (p? ) 2 normalized weight-q vectors of B(KjF ) with r = p s? +. Each of these produces a distinct blocking set of cardinality p s + p s? + in the projective translation plane. The following question arises: can there be a derivation set of cardinality p s? + in P G2(F p s)? Recall that we must have njdj vectors in D. Since n = q for translation planes, the \n" is given by the translates of a T. Thus we must nd jdj = p s? + subspaces, T0; T; : : : ; T p s?, with T m \ S i = f0g for any m implying T j \ S i = f0g for all j. Suppose we have found T0 with T0 \ S0 of dimension s? and T0 \ S i of dimension for i p s?. Since a k-dimensional subspace of an n-dimensional space over F q has qk(n? k) complementary (n?k)-dimensional subspaces, T0\S0 has q s? complementary subspaces

12 2 in S0. Next, suppose there is an s-dimensional subspace T with T \ S0 complementary to T0 \ S0, and T \ S complementary to T0 \ S. Then V = S0 S = (T0 \ S0 T \ S0) (T0 \ S T \ S) = (T0 \ S0 T0 \ S) (T \ S0 T \ S) T0 + T. Thus T0 \ T = f0g since both T0 and T are s-dimensional. It is thus conceivable that there could exist T0; T; : : : ; T p s? with T j \ S i = f0g for i > p s? and all j, and with T i \ S j of dimension s? for 0 i p s?, and T i \ T j = f0g for i 6= j. These subspaces and their translates would yield the desired q(p s? + ) vectors of D. Put in the language of spreads, T0; T; : : : ; T p s?, as a partial spread, would replace S0; S; : : : ; S p s?. For q = 2 3, the rst case possible, we have r = 5 and, because of the peculiar homogeneity properties of P? L2(8) acting on L, any 5-set of L would be suitable if one such could be found. One computes easily that there are seven T 's available for each 5-subset of L, but, unfortunately, ve of these seven cannot be chosen properly. It follows, since r + p 8, that the only non-trivial derivation sets have r = 7 and are given by ovals. We have not investigated any other case. We end by remarking that one could work in B(F q s jf q ), getting similar counts and possibilities. References. E.F. Assmus, Jr. and J.D. Key, Ane and projective planes, Discrete Math., Special Coding Theory Issue (to appear). 2. E.F. Assmus, Jr. and J.D. Key, Baer subplanes, ovals and unitals, Coding Theory, IMA Volumes in Mathematics and its Applications, ed. D. Ray-Chaudhuri, Springer, New York (989). 3. A.A. Bruen, Blocking sets in nite projective planes, SIAM J. Appl. Math., 2 (97), 380{ A.A. Bruen and R. Silverman, On extendable planes, M.D.S. codes and hyperovals in P G(2; q), q = 2 t, Geometriae Dedicata, 28 (988), 3{ P. Charpin, Codes cycliques etendus invariants sous le groupe ane, These de Doctorat d' Etat, Universite de Paris VII (987). 6. P. Delsarte, A geometric approach to a class of cyclic codes, J. Combin. Theory, 6 (969), 340{ P. Dembowski, Finite Geometries, Springer-Verlag, New York, U. Dempwol and A. Reifart, The classication of the translation planes of order 6, I, Geometriae Dedicata, 5 (983), 37{ N. Hamada, On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes, Hiroshima Math. J., 3 (973), J.W.P. Hirschfeld, Projective Geometries over Finite elds, Clarendon Press, Oxford, D.R. Hughes and F.C. Piper, Projective Planes, Springer Graduate Texts in Mathematics, New York, G. Kelly, Symmetric designs with translation blocks, Geometriae Dedicata, 5 (984), 233{ J.D. Key and K. Mackenzie, An upper bound for the p-rank of a translation plane, J. Combin. Theory A (to appear). 4. E. Kleinfeld, Techniques for enumerating Veblen-Wedderburn systems, J. Assoc. Comput. Mach., 7 (960), H. Luneburg, Charakterisierungen der endlichen desarguesschen projektiven Ebenen, Math. Z., 85 (964), 49{450.

13 3 6. H. Luneburg, Translation Planes, Springer-Verlag, New York, K. Mackenzie, Ph.D. thesis, University of Birmingham (989). 8. T.G. Ostrom, Semi-translation planes, Trans. Amer. Math. Soc., (964), {8. 9. T.G. Ostrom, Vector spaces and construction of nite projective planes, Archiv der Math., 4 (968), {25. E. F. Assmus, Jr. J. D. Key Department of Mathematics Department of Mathematical Sciences Lehigh University, Bldg. 4 Martin Hall, Clemson University Bethlehem, PA 805 Clemson, SC U.S.A. U.S.A.