Synopsis of the Ph.D. thesis Bounds and spectrum results in Galois geometry and coding theory Szabolcs Levente Fancsali Eötvös Loránd University Institute of Mathematics Doctoral School: Mathematics Director: Miklós Laczkovich Doctoral Program: Applied Mathematics Director: György Michaletzky Supervisor: Péter Sziklai Department of Computer Science Eötvös Loránd University 2011
Budapest, 29th May 2011
1 In my Ph.D. thesis, I discuss some spectrum results in Galois geometries. Here, spectrum usually means the set of possible sizes of a special set having a nice (or almost nice ) structure. Notation In my dissertation, p always denotes an arbitrary prime and q = p n (n 1) always denotes an arbitrary prime power (that can also be a prime). GF(q) denotes the finite field with q elements, and F can denote an arbitrary field (or also a Euclidean ring). PG(d, q) denotes the projective Galois geometry of dimension d over the finite field GF(q). AG(d,q) denotes the affine geometry of dimension d over GF(q) that corresponds to the co-ordinate space GF(q) d of rank d over GF(q). 1. Linear sets Definition (Affine linear set). A GF(s)-linear affine set is the GF(s)-linear span of some vectors in AG(n,q) = GF(s log s q ) n = GF(s) nlog s q (or possibly a translate of such a span). The rank of the affine linear set is the rank of this span over GF(s). The affine space AG(n,q) and its ideal hyperplane Π = PG(n 1,q) of directions together constitute a projective space PG(n,q). We say that the point P Π is a direction determined by the affine set U AG(n,q) if there exists at least one line through P that meets U in at least two points. Definition (Projective linear set). A projective GF(s)-linear set B of rank d + 1 is a projected image of the canonical subgeometry PG(d,s) PG(d,s n log s q ) from a center disjoint to this subgeometry. The projection can yield multiple points. Proposition 1.3. Suppose that U is an affine GF(s)-linear set of rank d+1 in AG(n,q) and let D denote the set of directions determined by U. The set U D is a projective GF(s)-linear set of rank d + 1 in PG(n,q) and all the multiple points are in D. The proposition above says that the set of directions determined by an affine linear set is a projective linear set. The converse of this proposition is also true: each projective linear set is a direction set. Theorem 1.6. Embed PG(n, q) into PG(n+1, q) as the ideal hyperplane and let AG(n+ 1,q) = PG(n+1,q) \ PG(n,q) denote the affine part. For each projective GF(s)-linear set D of rank d + 1 in PG(n,q), there exists an affine GF(q)-linear set U of rank d + 1 in AG(n+1,q) such that the set of directions determined by U is D. Definition (Club). Let L = PG(1,q h ) be a line of the projective plane PG(2,q h ), h 2. Let C denote a set of q 2 + 1 points in the line L. We say that C is a club of L if there exists a subplane = PG(2,q) of order q in PG(2,q h ), and there exists a point C on an extended line of but not in, such that C is the projected image of onto L from the center C.
2 By this projection, C has a special ( multiple ) point H called the head of the club, which is the projected image of the line of whose extension contains the center of projection. From now on, by a subline (without any attribute) we always mean a PG(1,q) contained in L. Definition (Regular and irregular sublines). A subline completely contained in the club C will be called regular (according to this particular construction) if it is the image of a line of. The sublines completely contained by the club C that cannot be got as an image of any subline of will be called irregular or deviant (according to this particular construction). Proposition 1.11. & 1.13. Let l be a subline of order q (and let an arbitrary construction of the club C be fixed). If the intersection l C contains the head of the club C and two other points of C, then l must be a regular subline of the club. If l C contains four (non-head) points, then l must be completely contained in the club C. If the club C is not equal to a subline PG(1,q 2 ) of order q 2 (that automatically holds when h is odd), then each subline completely contained by the club C is regular according to each possible construction of the club. If the club C equals to a subline PG(1,q 2 ) of order q 2 then the club, with the (regular and deviant) sublines it contains, forms a Moebius plane. I.e. for any 3 points of C there is a unique subline containing them, and this subline is contained in C as well. Corollary 1.14. A club and a subline intersect in 0, 1, 2, 3 or q + 1 points. 2. Directions determined by less than q points If U denotes an arbitrary set of points in the affine plane AG(2,q) then we say that the set { } b d D = a c (a,b), (c,d) U, (a,b) (c,d) is the set of directions determined by U. We define a 0 as if a 0, thus D l = GF(q) { }. Definition. If y D, then let s(y) denote the greatest power of p such that each line l of direction y meets U in zero modulo s(y) points. In other words, s(y) = gcd ({ l U l l = {y} } { p h}). Let s be the greatest power of p such that each line l of direction in D meets U in zero modulo s points. In other words, s = gcd s(y) = min s(y) y D y D Note that s(y) and thus also s might equal to 1.
3 Theorem 1.17 (Blokhuis, Ball, Brouwer, Storme and Szőnyi; Ball). Let U = q and / D. Using the notation s defined above, one of the followings holds: either s = 1 and q + 3 2 D q + 1; or GF(s) is a subfield of GF(q) and q s + 1 D q 1 s 1 ; or s = q and D = 1. Moreover, if s > 2 then U is a GF(s)-linear affine set (of rank log s q). Using the pigeon hole principle, one can easily prove that if U > q then it determines all the q+1 directions. So we can restrict our research to affine sets of less than q points. Notation. Let U be a set of less than q affine points in AG(2,q) and let D denote the set of directions determined by U. Let n = U and let R(X,Y ) be the inhomogeneous affine Rédei polynomial of the affine set U, that is, R(X,Y ) = (a,b) U n 1 (X + ay + b) = X n + σ n i (Y )X i where the abbreviation σ k (Y ) means the k-th elementary symmetric polynomial of the set {ay + b (a,b) U} of linear polynomials. Proposition 1.18. If y D then R(X,y) GF(q)[X s(y) ] \ GF(q)[X p s(y) ]. If y / D then R(X,y) X q X. Notation. Let F be the polynomial ring GF(q)[Y ] and consider R(X,Y ) as the element of the univariate polynomial ring F[X]. Divide X q X by R(X,Y ) as a univariate polynomial over F and let Q denote the quotient and let H + X be the negative of the remainder. So Q(X,Y ) = (X q X) div R(X,Y ) over F X H(X,Y ) (X q X) mod R(X,Y ) over F q 1 R(X,Y )Q(X,Y ) = X q + H(X,Y ) = X q + X q 1 i h i (Y ) where deg X H < deg X R. Let σ denote the coefficients of Q, and so Q(X,Y ) = X q n + h j (Y ) = q n 1 σ q n i(y )X i j σ i (Y )σj i(y )
4 Proposition 1.19. If y D then Q(X,y),H(X,y) GF(q)[X s(y) ] and if deg R deg Q then Q(X,y) GF(q)[X s(y) ] \ GF(q)[X p s(y) ]. If y / D then R(X,y)Q(X,y) = X q + H(X,y) = X q X. In this case Q(X,y) is also a totally reducible polynomial. Definition. Suppose that D 2. For each y D, let t(y) denote the maximal power of p such that H(X,y) = f y (X) t(y) for some f y (X) / GF(q)[X p ]. H(X,y) GF(q)[X t(y) ] \ GF(q)[X t(y)p ] Let t be the greatest common divisor of the numbers t(y). t = gcd t(y) = min t(y). y D y D If H(X,y) a (i.e. D = {y}) then we define t = t(y) = q. Proposition 1.20. Using the notation above, R(X,Y )Q(X,Y ) = X q + H(X,Y ) Span F 1,X,X t,x 2t,X 3t,...,X q. Theorem 1.26. Let U AG(2,q) be an arbitrary set of points and let D denote the directions determined by U. We use the notation s and t defined above geometrically and algebraically, respectively. Suppose that D. One of the followings holds: either D = { } and s t = q, U 1 U 1 or D > 1 and + 2 D t + 1 s 1 If s = 1 then the upper bound is q + 1. 3. Maximal partial 2-spreads and s t < q. A set of t-dimensional subspaces partitioning the points of PG(m,q) is called a t- spread. If t = 1 or t = 2 one may call it a line-spread or plane-spread, respectively. A partial t-spread in PG(m, q) is a set of pairwise disjoint t-dimensional subspaces. A partial t-spread is maximal if it is not contained in a larger partial t-spread. There exists a t-spread in PG(m,q) if and only if t+1 is a divisor of m+1. In this case one can define the deficiency of a partial t-spread of PG(m,q) which is the difference of the cardinalities of a t-spread and the partial t-spread considered. We constructed maximal partial plane-spreads of various deficiencies and we get the following results. Theorem 2.13. If q 16 then there exist maximal partial 2-spreads in PG(8,q) of deficiency δ = (k 1)q 2 where 1 k min { q 4 48 q 2 (log q + 1 4), q 2 + q + 1 }. Theorem 2.16. There exist maximal partial 2-spreads in PG(8, q) of deficiency δ = k q 2 + l (q 2 1) + 1, where k + l q 2.
5 Theorem 2.19. There exist maximal partial 2-spreads in PG(8, q) of deficiency δ = (k+1) q 2 + l (q 2 1) + m (q 2 2) + 1, where k + l + m q 2. Our fourth construction produces a partial 2-spread of various deficiencies. Theorem 2.23. In PG(3m 1,q), m 4, there are maximal partial 2-spreads of deficiency δ = ( x (k x 1) + y k y + z (k z + 1) ) q 2 + ( ) y l y + z l z (q 2 1) + z m z (q 2 2) + { y + z, where k x q 2 + q + 1, k y + l y } q 2, k z + l z + m z q 2 and x + y + z max q 2 q 3(m 2) 1, q2 3 (3m 7) log q. q 3 1 q 2 + 1 4. Multiple blocking sets and the Griesmer Bound A q-ary linear code C of length n, dimension k and minimum distance d, is a rank-k subspace of GF(q) n in which the Hamming distance between any two distinct vectors is at least d. If the dual minimum distance of the code C is at least three then the linear code C is called a projective code. Definition (Multiple blocking set). A t-fold blocking set with respect to the hyperplanes of PG(k 1,q) is a set of points B with the property that every hyperplane is incident with at least t points of B. A projective code of length n, dimension k and minimum distance d corresponds to a t-fold blocking set with respect to the hyperplanes of PG(k 1,q), where t = qk 1 1 q 1 n+d. Definition (Notation). Let q (t) denote the maximum number such that a t-fold blocking set in PG(2,q) has at least t(q + 1) + q (t) points. Griesmer bound. A k-dimensional projective code of minimum distance d has length n at least the Griesmer bound, k 1 d n g(k,d) := Definition (Bracket notation). If (s n,...,s 1 ) is an arbitrary n-tuple then let the expression θ q [s n,...,s 1 ] mean the following sum q i. θ q [s n,...,s 1 ] = n i=1 s i q i 1 q 1. Theorem 3.19. & 3.20. If B PG(δ,q) is a t = θ q [t δ 2,...,t 0 ]-fold blocking set with respect to hyperplanes such that there exists a t-secant hyperplane Π < PG(δ,q) then δ 2 B θ q [t δ 2,...,t 1,t 0, 0] = qt + t i
6 Let B PG(δ,q) be a t-fold blocking set with respect to hyperplanes such that there exists a t-secant hyperplane. If t δ 4,...,t 1,t 0 q 1 and t δ 2 1 and t δ 3 q (t δ 2 ) 1 then k 2 B θ q [t δ 2,...,t 1,t 0,q] = qt + t i + q. In what follows we assume that θ q [t δ 2,...,t 2,t 1 + 1] = t 1 + 1 and θ q [t δ 2,...,t 2 ] = 0 if δ = 3. Theorem 3.26. Let δ 3 and suppose that B PG(δ,p) is a t-fold blocking set with respect to hyperplanes and that t 0 1. If there exists a θ q [t δ 2,...,t 2,t 1 + 1]-secant (δ 2)-dimensional subspace, contained in a t-secant hyperplane but not contained in any hyperplane incident with at least p+t points of B, then δ 2 { } p + 1 B pt + t i + min 2,p t 0. Finally, we translate these results into their corresponding results in terms of linear codes meeting the Griesmer bound. Theorem 3.29. Let C be a k-dimensional linear code over GF(p) with minimum distance d < p k 1 which meets the Griesmer bound. Suppose there are codewords m 1,...,m m, where 2 m k 2, with the property that for j = 1,...,m 1 j j 1 supp(m i ) = d and If any of the following occurs, i=1 p i m m 1 supp(m i ) = d + 1. i=1 (1) p m 2 does not divide d and d m 2 max( p 1 2,p d m 1 1) and d m 2 p 1; (2) p m 2 does not divide d and d m 2 = p 1 and d m 1 p 1; (3) p m 2 divides d and d m 2 max( p+1 2,p d m 1); (4) p m 2 divides d and d m 2 = 0 and d m 1 0; then there is a codeword in Span m 1,...,m m of weight at least d + p. p i 5. Small weight codewords We define the incidence matrix A = [a ij ] of the projective plane PG(2,q), q = p h, p prime, h 1, as the matrix whose rows are indexed by lines of the plane and whose columns are indexed by points of the plane, and with entry a ij = { 1 if point j belongs to line i, 0 otherwise.
7 The p-ary code C of the projective plane PG(2,q), q = p h, p prime, h 1, is the GF(p)-span of the rows of the incidence matrix A. { } C = wa w GF(p) q2 +q+1 In [7], it is proven that the scalar multiples of the incidence vectors of the lines are the only codewords of minimal weight q + 1 in the code arising from PG(2,q). Kevin L. Chouinard [8] proved that for the code arising from PG(2,p), p prime, there are no codewords of weight in the interval [p + 2, 2p 1] and that the only codewords of weight 2p are the scalar multiples of the differences of the incidence vectors of two distinct lines. We prove the same result for q = p 3 where p is prime. Theorem 4.9. In the p-ary linear code of PG(2,p 3 ), p prime, p 7, there are no codewords with weight in the interval [p 3 + 2, 2p 3 1]. In general we can prove weaker results. Theorem 4.11. The p-ary linear code C arising from to the plane PG(2,q 3 0), q 0 = p h, p 7 prime, h 1, does not have codewords of weight q 3 0 + q 2 0 + 1 or of weight q 3 0 + q 2 0 + q 0 + 1; and if q 0 is a square, C has no codewords of weight q 3 0 + q 3/2 0 + 1. We know that a codeword m with weight in the interval [q + 2, 2q 1] defines a minimal blocking set of PG(2,q), q = p h, p prime, h 1, intersecting every line in 1 (mod p) points. We wish to exclude as many values as possible as weights for the codewords in the general case q = p h, with p prime, h 4. Consider a minimal blocking set B of size B < 2q in PG(2,q), q = p h, p prime, h 1, intersecting every line in 1 (mod p e ) points, with e the maximal integer for which this is true. Let p e = E. Theorem 4.13. There are no codewords with weight in [ 3 2 q, 2q 1] in the p-ary linear code of PG(2,q), q = p h, corresponding to a minimal blocking set intersecting every line in 1 (mod E) points when E = p e 4. Theorem 4.14. When B is a minimal blocking set in PG(2,q = p h ), p prime, h 1, of size B 2q 1, intersecting every line in 1 (mod p e ) points with e the maximal integer for which this is valid, then for large prime numbers p, q B q + a 0 p + a q e 1 p + + a h/e 2p e + 1, 2e with a i the i-th Motzkin number. 6. Network coding A communication network consists of a finite directed graph G = (V, E) with source node s V and sink nodes t 1,...,t k V, such that information (one element of a given field GF(q) in every time step) can be sent noiselessly via all e E. Our aim is to
8 send a message m GF(q) d of length d via the network. The multicast network coding problem for a given communication network and message length is to give functions f e : GF(q) (e) GF(q) for all edge e E (where (e) denotes the in-degree of the tail of the edge e), such that the character sent on the edge e = (u,v) is the value of f e applied to the incoming characters of u = Tail(e), and every sink node t i must be able to recover the original message m from its incoming characters. Definition (Linear network). Let G = (V, E) be a directed graph and let Φ : E U be a function such that each co-vector Φ(e) U is the linear combination of the co-vectors {Φ(f) f E : Head(f) = Tail(e)} and for each cut C E the set {Φ(e) e C} U is a generator system of U. Then the pair (G, Φ) is a linear network. Let (G, Φ) be a linear network. Consider the problem of security against a wire-tap adversary who can eavesdrop k fixed edges. The idea of secure network coding is that the source mixes the original message with some random noise, such that the mutual information between the original message and the eavesdropped characters is zero. Adversaries. The adversary who eavesdrops the edges {e 1,...,e k } E can be represented by the linear operator A = [ Φ(e 1 ),...,Φ(e k ) ] Hom ( U, GF(q) k). Adversary A knows exactly that in which translate of kera = {Φ(e 1 ),...,Φ(e k )} U the message m U lies. Source process. Let Σ be a finite alphabet (or vocabulary). The source stochastic process is a series X = ( X 1,...,X T ) of Σ-valued probability variables. Our aim is to give a correspondence between Σ and U such that certain wire-tap adversaries cannot get any information about the source process; but each sink node can recover the whole original source process. Encryption and decryption. Let be another finite alphabet (or vocabulary) and let the random noise be the stochastic process W = ( W 1,...,W T ) of -valued probability variables. Definition (Encryption). The pair of functions Enc : Σ U and Dec : U Σ is called an encryption system or briefly an encryption from Σ to U, if Dec(Enc(σ, δ)) = σ. If the mutual information I ( Y(A);X ) = 0 for each matrix A consisting of k elements of RanΦ then the encryption system is called a k-encryption. Corollary 5.8. Let K = min{ ker A A A}. If the encryption system makes the source process secure against each adversary in A, then Σ K. We know that K = q d k where k = max{ranka A A} and d = rank U. Thus, the inequality above yields the upper bound m = log q Σ d k.
9 Definition. We call an encryption optimal if Σ = K; (using the notation above) let the number δ = rank U log q Σ k = d m k be called the deficiency of a k-encryption (of Σ into U). Definition (Linear encryption). If the encryption function is linear in its first variable then the encryption is called linear encryption. If Σ can be coordinatized (or recoordinatized ) such that the encryption becomes linear then we say that this encryption is an essentially linear encryption. Of course, a linear encryption is essentially linear as well. Definition. A universal k-encryption of Σ into U is a pair of an encryption and decryption function such that for every linear network coding Φ : E U and for every adversary A, with ranka = k, the mutual information I ( Y(A);X ) is zero. Theorem 5.15. Let G = (V,E) be a network and let U = GF(q) d be a vector space such that there exist linear network codings for G using U. For any Σ ( Σ 2) and for any k {1,..., d 1}, there does not exist universal k-encryption from Σ to U. Now suppose that both the communication network G = (V,E) and the linear coding Φ : E U are given. Let δ k denote the minimal deficiency of all the uniform linear k-encryptions in the given linear network (G, Φ). Since the deficiency δ of a k-encryption is less than rank U k, let the cases δ k d k mean linear k-encryption does not exist. Let K k denote K k = Span(f1,...,f k). f1,...,f k RanΦ Let the linear space U coordinatize the projective space PG(d 1,q) and let K k be considered as a projective point set in PG(d 1,q), so K 1 is the projective point set coordinatized by RanΦ. Let R = {P(f ) f RanΦ} = K 1, where P(f ) denotes the projective point coordinatized by the vector f. Theorem 5.22. The minimal possible deficiency δ k is equal to d k r if and only if K k meets each projective subspace of dimension r but there exists at least one projective subspace of dimension (r 1) disjoint from K k. A subset H of points of PG(d 1,q) is called a blocking set with respect to hyperplanes iff H meets every hyperplane of PG(d 1,q) in at least one point. Proposition 5.23. There exists an optimal linear 1-encryption using uniform noise if and only if the projective point set coordinatized by RanΦ U is not a blocking set w.r.t. projective hyperplane of U. Proposition 5.24. If there exists linear k-encryption of deficiency δ for k 2 then R qk+δ 1 q 1.
10 Corollary 5.25. If there exists optimal linear k-encryption for k 2, then R qk 1 q 1. Remember that K k is a generator system and is a union of some projective subspaces of dimension k 1. Moreover, K k is a blocking set with respect to projective subspaces of dimension d k δ k. Sometimes we can suppose that K k does not contain projective subspace of dimension k. These facts explain the following Proposition 5.29. Let s = δ+1 k. If ( R k) q δ s k+1 s j=0 qj k then there exists a uniform linear k-encryption of deficiency δ, especially if ( R k) q then there exists an optimal uniform linear k-encryption. As a consequence of this proposition we can get a bound for the size of the field similar to Feldman et al. [9]. Corollary 5.30. If R k δ+1 < q then there exists a uniform linear k-encryption of deficiency δ. Proposition 5.31. If k j=0 ( R j) (q 1) j < q k+1 then there exists an optimal linear k-encryption. Corollary 5.32. ( R k) k + 1 < q then there exists an optimal uniform linear k- encryption. The Ph.D. thesis is based on [1] S. Ball and Sz. L. Fancsali Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes. Des. Codes Cryptogr. 53, 2 (2009), 119 136. [2] V. Fack, Sz. L. Fancsali, L. Storme, G. Van de Voorde and J. Winne Small weight codewords in the codes arising from Desarguesian projective planes. Des. Codes Cryptogr. 46, 1 (2008), 25 43. [3] Sz. L. Fancsali and P. Ligeti Some applications of finite geometry for secure network coding. J. Math. Cryptol. 2, 3 (2008), 209 225. [4] Sz. L. Fancsali and P. Sziklai About maximal partial 2-spreads in PG(3m 1,q). Innov. Incidence Geom. 4 (2006), 89 102. [5] Sz. L. Fancsali and P. Sziklai Description of the clubs. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 51 (2008), 141 146 (2009). [6] Sz. L. Fancsali, P. Sziklai and M. Takáts The number of directions determined by less than q points. Submitted to the J. Algebraic Combin. Further references [7] E. F. Assmus, Jr. and J. D. Key Designs and their codes, vol. 103 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1992. [8] K. L. Chouinard Weight distributions of codes from planes. Ph.D. thesis, University of Virginia, 2000. [9] J. T. M. Feldman, C. Stein and R. A. Servedio On the capacity of secure network coding. In Proc. 42nd Annual Allerton Conference on Communication, Control and Computing (2004).