Journal of Symbolic Computation. On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

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1 Journal of Symbolic Computation 47 (2012) Contents lists available at SciVerse ScienceDirect Journal of Symbolic Computation journal homepage: wwwelseviercom/locate/jsc On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field Matthew T Comer, Erich Kaltofen Department of Mathematics, North Carolina State University, Raleigh, NC , USA a r t i c l e i n f o a b s t r a c t Article history: Received 14 October 2010 Accepted 25 February 2011 Available online 16 September 2011 Keywords: Toeplitz matrix Hankel matrix Block matrix Finite field Singularity counts Fixed entry Berlekamp/Massey algorithm We derive an explicit count for the number of singular n n Hankel (Toeplitz) matrices whose entries range over a finite field with q elements by observing the execution of the Berlekamp/Massey algorithm on its elements Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed For example, the number of singular n n Toeplitz matrices with 0 s on the diagonal is q 2n 3 + q n 1 q n 2 We also derive the count for all n n Hankel matrices of rank r with generic rank profile, ie, whose first r leading principal submatrices are non-singular and the rest are singular, namely q r (q 1) r in the case r < n and q r 1 (q 1) r in the case r = n This result generalizes to block-hankel matrices as well 2011 Elsevier td All rights reserved 1 Introduction Throughout the discussion, entries will be taken from the field F q of q elements, and we will identify a square Hankel matrix a 0 a 1 a n 2 a n 1 a 1 a 2 a n 1 a n a n 2 a n 1 a 2n 4 a 2n 3 = [a i+j 2] n i,j=1 a n 1 a n a 2n 3 a 2n 2 with the list [a 0, a 1,, a 2n 2 ] A Toeplitz matrix is the mirror image [a n+i j 1 ] n i,j=1 This material is based on work supported in part by the National Science Foundation under Grant CCF addresses: mcomer@ncsuedu (MT Comer), kaltofen@mathncsuedu (E Kaltofen) UR: (E Kaltofen) /$ see front matter 2011 Elsevier td All rights reserved doi:101016/jjsc

2 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Our investigation was motivated by the question of Ramamohan Paturi who asked in October 2009 how many Toeplitz matrices over F q with zeros on the main diagonal were non-singular Paturi needed the estimate for the complexity of circuit satisfiability for lower bounds (Paturi and Pudlák, 2010) Daykin (1960) proved theorems regarding the number of Hankel matrices over a finite field with a specified rank or determinant Kaltofen and obo (1996) established some of Daykin s counts using the extended Euclidean algorithm form of the Berlekamp/Massey algorithm for polynomials (Sugiyama et al, 1975) Additionally, they gave the number of square Toeplitz matrices with generic rank profile Generic rank profile means that for a matrix A of rank r, the first r leading principal submatrices A 1,, A r are non-singular We prove analogous results here, albeit with a different approach, for the Hankel case The counts for Toeplitz and Hankel matrices of generic rank profile are not the same The determination of singularity of a Hankel matrix has a natural connection with running the Berlekamp/Massey algorithm on the list [a 0, a 1,, a 2n 2 ], and for this reason we count how many Hankel matrices have zeros along the anti-diagonal in order to answer the question regarding Toeplitz matrices Kaltofen and ee (2003) have observed that the Berlekamp/Massey algorithm (Massey, 1969, cf Theorem 1) detects the non-singular leading principal submatrices of a Hankel matrix from those non-zero discrepancies that increase the linear generator degrees, and that the corresponding sequence elements determine the singularity of the corresponding leading principal submatrices We will use this property to partition the space of Hankel matrices into unique correspondences of one singular Hankel matrix to q 1 non-singular Hankel matrices This process generalizes when particular entries of the list [a 0,, a 2n 2 ] are fixed to arbitrary values (such as the case of zeros along the anti-diagonal) We then investigate the properties of block-hankel matrices We have no explicit formula for how many block-hankel matrices are singular (with or without certain blocks fixed), but we present some brute-force counts that we have computed with Maple ast, we follow in the theme of (Kaltofen and obo, 1996) by counting block-hankel matrices with block generic rank profile A block matrix A (of square submatrices of dimension m) of rank mr has block generic rank profile if for k = 1, 2,, r, we have rank(a k ) = mk, where A k is the k k block leading principal submatrix of A The counts for unblocked rank r Hankel matrices are given in (Garcìa-Armas et al, 2011) 2 Connection with the Berlekamp/Massey algorithm Given a list of n field elements, [a 0,, a n 1 ], the Berlekamp/Massey algorithm will produce for each r {1, 2,, n} a monic polynomial Λ r = c 0 + c 1 z + + c r 1z r 1 + z r of minimal degree r r 1 such that c 0 a i c 1 a i+1 c r 1a i+r 1 = a i+r, i = 0, 1,, r r 1 (1) Such a polynomial is called a (minimal) generating polynomial, and r is called the minimal length required to generate the first r elements of the sequence Theorem 2 in (Massey, 1969) states that if a polynomial (of minimal degree n ) generates the list [a 0,, a n 1 ] but fails to generate a n (ie, Eq (1) does not hold for i = n r ), then the minimal generating polynomial of [a 0,, a n ] will be of degree n+1 = n n + 1 We visualize the generating polynomials in the following way: for each r = 1, 2,, n, define a 0 a 1 a r 2 a r 1 a 1 a 2 a r 1 a r H r = a r r 1 a r r a r 1 a r 2, λ r = a r r a r r +1 a r 2 a r 1 c 0 c 1 c r 2 c r 1, h r = a r a r +1 a r 1 a r so that by definition of Λ r above, we have H r λ r = h r (for r = 1, 2,, n 1) except possibly in the last entry (which corresponds to Eq (1) for i = r r ) The last entry of H r λ r will not be a r if and only if Λ r generates [a 0,, a r 1 ] but not a r,

3 482 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Suppose that Λ n of degree n n generates [a 0,, a n 1 ] but not a n Then the vector Λ n = [ λ n 1 ] T = [ c n c n 1 c 1 1 ] T is a null-space vector of the leading (n n ) ( n + 1) submatrix of the matrix depicted in Fig 1; the first n columns of this submatrix form the first n n rows of H n 1 As noted above, Theorem 2 in (Massey, 1969) implies that the minimal length required to generate [a 0,, a n ] is n+1 = n n + 1 Thus, H n+1 will have n n + 1 columns, and in fact, the entire (n n +1) (n n +1) leading principal submatrix in Fig 1 will be non-singular For completeness, we now give those details in the proof of emma 2 (respectively, emma 17) in (Kaltofen and Yuhasz, 2006; Yuhasz, 2009), which justify that claim If we post-multiply the (n n + 1) (n n + 1) leading principal submatrix a 0 a n 1 a n n a n 1 a 2(n 1) a n 1 a n n a n 1 a 2(n n ) by I n 0 m 1 m 2 c n c n 1 c n c n 1 0 c 1 cn 0 1 c 1 cn 1 c n, 0 1 cn 1 0 c1 1 c m 1 = n 2 n + 1 m 2 = n, (2) the result is H 0 m 1 m 2 * 0 0 α α 0 0 α α, m 1 = n, α 0, (3) m 2 = n 2 n + 1 where H is the n n leading principal submatrix of H and α 0 is the last entry of H n λ n The (n n + 1) (n n + 1) leading principal submatrix of H will be non-singular if the above matrix is non-singular, which will happen if H is non-singular This can be shown as a consequence of emmas 1 and 2 below emma 1 Given a list [0, 0,, 0, α] with α 0 as the (k + 1)-st entry, the first H r with r > 0 will be H k+1 = [0 0 0 α] Proof The Berlekamp/Massey algorithm initializes Λ 0 = 1, which generates the zero sequence of any length Thus we have Λ 1 = = Λ k = 1 and 0 = = k = 0, where Λ k generates a 0, a 1,, a k 1 (the zero sequence) but not a k = α We then have

4 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Fig 1 Berlekamp/Massey algorithm, as seen in (Kaltofen and ee, 2003) k+1 = max{ k, k k + 1} = max{0, k 0 + 1} = k + 1, and Λ k+1 = z k+1 So for r k, H r is of size (r r + 1) r = (r + 1) 0 (ie, a matrix of r + 1 empty rows), and H k+1 is of size (k + 1 k+1 + 1) k+1 = (k + 1 (k + 1) + 1) (k + 1) = 1 (k + 1), and thus H k+1 has the proposed form With H k+1 as in emma 1, the H r matrices will keep adding rows until a length change (in the minimal generator) occurs We can show that a length change will not occur until the H r matrices fill-up the rest of the rows of the k+1 k+1 leading principal submatrix of H emma 2 Suppose H p is a leading submatrix of rows of the p p leading principal submatrix of H, and suppose p = p+1 = = p+q < p+q+1 Then H p+q will have at least p + 1 rows, and H p+q+1 will have exactly p + 1 rows Proof We see that H p, H p+1,, H p+q all have the same number of columns, so by the definition of H r for arbitrary r, these matrices will be formed by augmenting by one row at a time Also by the definition of H r, we have a 0 a p+q 1 a 0 a p 1 H p+q = = a p+q p+q a p+q 1 a p+q p a p+q 1

5 484 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Because of the length change between p+q and p+q+1, we will have p+q+1 = max{ p+q, (p + q) p+q + 1} = (p + q) p+q + 1 = p + q p + 1 > p+q = p, so that H p+q will have more than p rows Also, we will have a 0 a p+q+1 1 a 0 a p+q p H p+q+1 = =, a p+q+1 p+q+1 a p+q a p a p+q so that H p+q+1 will have p + 1 rows Corollary 3 If r+1 > r, then H r+1 is a leading submatrix of rows of the r+1 r+1 leading principal submatrix of H From emma 1, it is clear that the first p p leading principal submatrix of H is non-singular (for p > 0) From emma 2, we see that when the next length change occurs, say p+q+1 > p, H p+q will have at least p + 1 rows, which corresponds to Λ p generating [a 0,, a p+q 1 ] but not a p+q We can post-multiply the p+q p+q leading principal submatrix by an appropriate matrix like (2) to obtain a matrix product whose result is analogous to (3) Because H here is the p p non-singular leading principal submatrix of H, we have that the p+q p+q leading principal submatrix of H is non-singular Using induction on emma 2, we conclude that for all n, the n n leading principal submatrix is non-singular (when n > 0) emma 4 Given an n n Hankel matrix H, let r be maximal such that r r + 1 n and r n (ie, H r is a submatrix of H but H r+1 is not) Then H r+1 will have n + 1 rows and at most n columns Proof We make the convention that if s = 0 for some s, then H s has s s + 1 = s + 1 empty rows, and is (trivially) a submatrix of H If r+1 = r n, then we will have a 0 a 1 a r+1 1 a 0 a 1 a r 1 a 1 a 2 a r+1 a 1 a 2 a r H r+1 = a r r+1 a r 1 = a r r a r 1 a r+1 r+1 a r H r =, a r+1 r a r+1 (r 1) a r a r+1 r a r so that H r+1 has r+1 = r n columns By the maximality of r, we must have n < (r + 1) r = (r r + 1) + 1 n + 1, so H r+1 has n + 1 rows If r+1 > r, then by Theorem 2 in (Massey, 1969) we have r+1 = r r + 1 n, so H r+1 has at most n columns Again by the maximality of r, we must have n < (r + 1) r = (r + 1) (r r + 1) + 1 = r + 1 n + 1, so again H r+1 has n + 1 rows emma 4 implies the following for any n n Hankel matrix H: if we run the Berlekamp/Massey algorithm on the entries of H, then there will be an r {1,, 2n 2} such that H r is a leading submatrix of entire columns of H, and H r+1 is obtained by augmenting an appropriate row of entries to H r, but H r+1 is not a submatrix of H

6 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Definition 5 We will say that the Berlekamp/Massey algorithm exits an n n Hankel matrix H at r if H r is a submatrix of H but H r+1 is not We make the convention that if 1 = = n = 0 (so that H n 1 is n 0 and H n is (n + 1) 0), then we say that the algorithm exits at n 1 Also, we use the terminology exits at 2n 1 even though a 2n 1 is not defined in H emma 6 et H be a square Hankel matrix If A is a non-singular leading principal submatrix of H, then H r = A for some r 1, when running the Berlekamp/Massey algorithm on the entries of H Proof et a 0 a k 1 A = a k 1 a 2k 2 Then because A is a square Hankel matrix, emma 4 implies that the Berlekamp/Massey algorithm will exit A at one of k 1, k,, 2k 1 et m 1 be the index and suppose m 1 2k 2 Then we may write H m = à y T ã, h m = α (m m +1) m, (m m +1) 1 where à is an appropriate leading submatrix of columns of A, and ã is an appropriate column of A Then we have A = à ã B, so that à λm ã B 1 0 = 0, hence A is singular, a contradiction Thus, we must have that the Berlekamp/Massey algorithm exits A at 2k 1, so that H2k 1 A H 2k = y T = y T, hence H 2k 1 = A Corollary 7 The Berlekamp/Massey algorithm will exit an n n Hankel matrix H at one of n 1, n,, 2n 2 if and only if the matrix is singular; the algorithm will exit at 2n 1 if and only if the matrix is non-singular Proof By the proof of emma 6, the algorithm will exit at 2n 1 if H is non-singular Now suppose that the algorithm exits at 2n 1 Then H 2n will be formed by adding a row to H 2n 1, which will be H itself, and we have H2n 1 H H 2n = y T = y T, so that a 0 a 2n 1 1 H = H 2n 1 = a 2n 1 2n 1 a 2n 2 Then we see that 2n 1 = n, hence H is 2n 1 2n 1, and thus non-singular by the proof of emma 2 (respectively, emma 17) in (Kaltofen and Yuhasz, 2006; Yuhasz, 2009)

7 486 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Counting singular Hankel matrices and et H n n denote the set of all n n Hankel matrices We define maps ϕ : H n n non-singular H n n singular [a 0,, a k,, a 2n 2 ] [a 0,, a k 1, a k, a k+1,, a 2n 2 ] ψ : H n n n n singular P (H ) non-singular [a 0,, a k,, a 2n 2 ] [a 0,, a k 1, a, k a k+1,, a 2n 2 ] a k F q \ {a k } in the following way: we run the Berlekamp/Massey algorithm on the list of entries associated with a Hankel matrix H, and we let k be maximal such that k < n and k k + 1 = n (ie, H k is a proper submatrix of columns of H) Because k < n, H k+1 will exit H if and only if H is singular; H k+1 will have k+1 > k columns (and remain a submatrix of H) if and only if H is non-singular We may factor H as a 0 a k 1 a k Hk h k B = B a k k a k 1 a k = y T 0 a k B y T k k a k where B may be empty, and H k λ k h k = (0, 0,, 0, y T k k λ k a k ) T We will have H non-singular if and only if y T k k λ k a k, again by the proof of emma 2 (respectively, emma 17) in (Kaltofen and Yuhasz, 2006; Yuhasz, 2009), so we define ϕ(h) = [a 0,, a k 1, y T k k λ k, a k+1,, a 2n 2 ] Similarly, if H is singular (so that y T k k λ k = a k ), then changing a k to any other value will result in a non-singular matrix, so we define ψ(h) = [a 0,, a k 1, a k, a k+1,, a 2n 2 ] a k a k emma 8 Given H H n n singular, ϕ ψ(h) = H Proof Given H H n n singular, say H = [a 0,, a 2n 2 ], we run the Berlekamp/Massey algorithm on the list of entries of H to get ψ(h) = [a 0,, a k 1, a k, a k+1,, a 2n 2 ] a k a k If we run the Berlekamp/Massey algorithm on ψ(h), then H i, h i and λ i will agree for i = 1, 2,, k, and we will have y T k k λ k = a k a k By the discussion of ϕ above, we will have as proposed ϕ ψ(h) = [a 0,, a k 1, y T k k λ k, a k+1,, a 2n 2 ] = [a 0,, a k 1, a k, a k+1,, a 2n 2 ] = H emma 8 immediately implies that if H H in H n n singular, then ψ(h) ψ( H) = { }: if a matrix H were in the intersection, then we would have H = ϕ( H) = H, a contradiction We now use these maps to count singular n n Hankel matrices, allowing a collection of entries to be fixed to prescribed values

8 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Definition 9 Given an n n Hankel matrix [a 0,, a 2n 2 ], we define = ( ind, val ) = ([i 1,, i k ], [α 1,, α k ]) (where k 2n 1) to represent fixed entries in H, where a ij is fixed to α j When counting singular Hankel matrices over F q, we let a i vary over F q if i ind We will let H n n denote the set of Hankel matrices with entries fixed according to Note that card(h n n ) = q 2n 1 k Theorem 10 (General Count) The number of singular n n Hankel matrices with entries fixed according to (as in Definition 9), where either ind {0,, n 1} or ind {n 1,, 2n 2}, is equal to q 2n 2 k, if n 1 ind or if n 1 ind with some σ (n, q, ) = other j ind and α j 0 q 2n 2 k q n 2, if n 1 ind, α n 1 0, and all other α j = 0 q 2n 2 k q n 2 + q n 1, if n 1 ind, α n 1 = 0, and all other α j = 0 Proof We first prove the counts for ind {0,, n 1} Suppose that n 1 ind Given a singular Hankel matrix H H n n, we run the Berlekamp/ Massey algorithm on the entries of H; the algorithm will exit at one of the entries on the bottom row because H is singular Note that n 1 ind implies that ϕ 1 (H) will yield a unique set of q 1 nonsingular Hankel matrices for every singular H H n n (because there is no restriction on any entry of the bottom row) It follows that a fraction of 1/q of the q 2n 1 k matrices in H n n will be singular Next, suppose that n 1 ind and α j 0 for some other j ind We again run the Berlekamp/ Massey algorithm on the entries of a singular H H n n, but now we have a restriction on an element of the bottom row (ie, a n 1 ), which poses a problem if the algorithm exits at a n 1 However, the condition α j 0 for some j {0,, n 2} guarantees that the largest proper non-singular leading principal submatrix will have size at least 1 1, so the algorithm cannot exit at a n 1 Thus, the map ϕ is defined for each non-singular H H n n and is surjective, so the count follows as in the case of n 1 ind Now suppose that n 1 ind, α n 1 0, and all other α j val are zero Consider the subset N of H n n where a 0 = = a n 2 = 0; there are q n 1 such matrices, which are all non-singular because α n 1 0 The map ϕ is not defined on this set because the Berlekamp/Massey algorithm would exit at a n 1, and thus ϕ would attempt to change a n 1 to zero, which cannot be done The matrices in N do not contribute to the count, so we restrict the domain of ϕ to (H n n H n n non-singular ) \ N As above, the condition α j 0 for some j {0,, n 2} guarantees that the map ϕ is defined for each nonsingular matrix in H n n \N, and again is surjective It follows that a fraction of 1/q of the q 2n 1 k q n 1 matrices in H n n \ N is singular ast, suppose that n 1 ind, α n 1 = 0, and all other α j val are zero We consider the set N from above, but now all matrices in N are singular because α n 1 = 0 Restricting ϕ to H n n ) \ non-singular N yields the same result, so we simply add the qn 1 singular matrices in N to (H n n the previous count To prove the result for ind {n 1,, 2n 2}, let J n = Fn n q be the anti-identity matrix Consider the linear transformation T J : H n n H n n via H J n HJ 1 n, which is a bijection onto its image, where is the set obtained by mapping each j ind to 2n 2 j

9 488 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Note that in H n n, all fixed entries are along or above the anti-diagonal, as they were in the case of ind {0,, n 1} We can therefore use the methods above to conclude the same counts in H n n Because T J is a bijection onto its image (and preserves singularity/non-singularity), we conclude the same counts for H n n 4 Counting block-hankel matrices with block generic rank profile Definition 11 We say that a block matrix A (of square submatrices of dimension m) of rank mr has block generic rank profile if rank(a k ) = mk for k = 1, 2,, r, where A k is the k k block leading principal submatrix of A emma 12 The number of block-hankel matrices (m m submatrices arranged in r r block-hankel form) of rank mr with block generic rank profile, denoted by H mr mr bgrp (r), is q m2 (r 1) m 1 i=0 (qm q i ) r Proof The proof follows by induction For the case r = 1, H m m blps is simply the number of non-singular m m matrices over F m 1 q, which is i=0 (qm q i ) Now let r > 1 and suppose that the (r 1) (r 1) block leading principal submatrix A r 1 is non-singular: H = M 0 M 1 M r 2 M r 1 M 1 M 2 M r 1 M r M r 2 M r 1 M 2r 4 M 2r 3 M r 1 M r M 2r 3 M 2r 2 = Ar 1 C r 1 B r 1 M 2r 2 We derive conditions on M 2r 3 and M 2r 2 that make H non-singular It is clear that for any choice of M 2r 3, the system A r 1 X = M 2r 3 has a unique solution We now determine conditions on the columns of M 2r 2 et the columns of B r 1 be denoted b 0, b 1,, b m 1, and the columns of M 2r 2 denoted v 0, v 1,, v m 1, and consider the matrix Ar 1 b 0 v 0 C r 1 The system A r 1 x = b 0 will have a unique solution x regardless of b 0, and correspondingly the block 2 2 matrix above will have full column rank if and only if C r 1 x v 0 Thus, there are (q m 1)-many choices for v 0 Next, suppose that we have chosen v 0,, v t 1 so that the matrix Ar 1 b 0 b t 1 C r 1 v 0 v t 1 has full column rank Then the matrix Ar 1 b 0 b t 1 b t C r 1 v 0 v t 1 v t will have full column rank if and only if the vector (b t, v t ) T is not in the span of the previous columns We see that if the system Ar 1 bt + t 1 x = α i=0 ib i C r 1 v t + t 1 α i=0 iv i has a solution, then it will be unique by the non-singularity of A r 1 Thus, for each choice of (α 0,, α t 1 ) F t q, there is one vector that v t must avoid, and so there are (q m q t )-many choices for v t It follows that the number of suitable matrices M 2r 2 is m 1 i=0 (qm q i )

10 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) Combining this with the fact that M 2r 3 may be arbitrary, we see that the number of block-hankel matrices (with A r 1 as the mr mr block leading principal submatrix) that have every block leading principal submatrix non-singular is q m2 m 1 i=0 (qm q i ) Overall, it follows that there are q m2 (r 1) m 1 i=0 (qm q i ) r block-hankel matrices of rank mr with block generic rank profile Theorem 13 The number of block-hankel matrices (m m submatrices arranged in n n block-hankel form) of rank mr with block generic rank profile, denoted by H mn mn bgrp (r), is equal to m 1 r, q m2 r (q m q ) i r < n H mn mn i=0 bgrp (r) = m 1 r, q m2 (r 1) (q m q ) i r = n i=0 Proof The case r = n is proved in emma 12, so we assume r < n et H be such a matrix Then we can write M 0 M r 1 M r M r+1 M n 1 M H = r 1 M 2r 2 M 2r 1 M 2r M n+r 2 M r M 2r 1 M 2r M 2r+1 M n+r 1 M n 1 M n+r 2 M n+r 1 M n+r M 2n 2 = M 2r D A r B r C r+1 C n M n+r 1, where A r is non-singular It is clear that for any choice of M 2r 1, the system A r X = B r has a unique solution et the columns of B r be denoted b 0, b 1,, b m 1, and similarly let the columns of [M T 2r M T n+r 1 ]T be denoted v 0, v 1,, v m 1 As in the proof of emma 12, consider the matrix Ar b 0 v 0 C r+1 The system A r x = b 0 will have a unique solution x regardless of b 0, and correspondingly the block 2 2 matrix above will have rank mr if and only if C r+1 x = v 0 The matrix H must have rank mr, so we see that v 0 is predetermined Next, suppose that Ar b 0 b t 1 C r+1 v 0 v t 1 has rank mr Then the matrix Ar b 0 b t 1 b t C r+1 v 0 v t 1 v t will have rank mr if and only if the vector (b t, v t ) T is in the span of the previous columns, which is equivalent to (b t, v t ) T being in the span of the first mr columns Given any b t, we see that the system Ar bt x = C r+1 v t

11 490 MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) will have at most one solution, by the non-singularity of A r There is a unique choice of v t that will make the system solvable: we set v t to C r+1 x, where x is the unique solution to A r x = b t Thus, v t is predetermined It follows that M 2r,, M n+r 1 are predetermined Moreover, M n+r,, M 2n 2 are predetermined, a fact that was corrected by an anonymous referee To see why, note that the first mr columns of H form a basis for the column space of H If we denote the columns of D by d 0, d 1,, d n r 1, then the matrix M 2r d 0 A r B r C r+1 C n M n+r 1 will have rank mr if and only if d 0 is in the span of the first m(r + 1) columns of H, which equals the span of the first mr columns of H By the non-singularity of A r, the resulting column relation would be unique, so the last m entries of d 0 (ie, the first column of M n+r ) are predetermined The same argument follows inductively for every column of D, so that each of M n+r,, M 2n 2 is predetermined Because only M 2r 1 may be arbitrary, it follows that H is one of m 1 r m 1 r q m2 (r 1) (q m q i ) q m2 = q r m2 (q m q i ) many matrices i=0 i=0 For the case m = 1, Theorem 13 implies that the number of n n Hankel matrices (with entries from F q ) of rank r with generic rank profile is (r) = qr (q 1) r, r < n bgrp q r 1 (q 1) r, r = n H n n We can compare this to the result in (Kaltofen and obo, 1996), which states that the number of n n Toeplitz matrices (with entries from F q ) of rank r with generic rank profile is 2 r 1 q 1 2n q 1, 0 < r < n q q N r = 2 n 1 q 1 2n q 1, r = n q q 2 We have investigated the properties of block-hankel matrices, but we do not know explicit formulas for how many block-hankel matrices are singular (with or without certain blocks fixed) Presented below are some brute-force counts for the number of singular block-hankel matrices (m m submatrices arranged in n n block-form, with entries from F q ) m n q Singular/Total = = = = = =

12 Acknowledgements MT Comer, E Kaltofen / Journal of Symbolic Computation 47 (2012) The authors would like to thank the referees for their comments, which improved the presentation of the paper References Daykin, DE, 1960 Distribution of bordered persymmetric matrices in a finite field J Reine Angew Math 203, Garcìa-Armas, M, Ghorpade, S R, Ram, S, 2011 Relatively prime polynomials and nonsingular hankel matrices over finite fields J Combin Theory Ser A 118, Kaltofen, E, ee, W, 2003 Early termination in sparse interpolation algorithms In: Giusti, M, Pardo, M (Eds), Internat Symp Symbolic Algebraic Comput (ISSAC 2002) J Symbolic Comput 36 (3 4), (special issue) EKbib/03/K03pdf Kaltofen, E, obo, A, 1996 On rank properties of Toeplitz matrices over finite fields In: akshman, YN (Ed), Proc 1996 Internat Symp Symbolic Algebraic Comput ISSAC 96 ACM Press, New York, NY, pp UR: EKbib/96/Kao96_issacpdf Kaltofen, E, Yuhasz, G, 2006 On the matrix Berlekamp Massey algorithm Manuscript, 29 pages (submitted for publication) Massey, J, 1969 Shift-register synthesis and BCH decoding IEEE Trans Inf Theory it-15, Paturi, R, Pudlák, P, 2010 On the complexity of circuit satisfiability In: Schulman, J (Ed), STOC, ACM, pp Sugiyama, Y, Kasahara, M, Hirasawa, S, Namekawa, T, 1975 A method for solving key equation for decoding Goppa codes Information & Control 27, Yuhasz, G, May 2009 Berlekamp/Massey algorithms for linearly generated matrix sequences PhD Thesis, North Carolina State Univ, Raleigh, North Carolina