Codes and Designs in the Grassmann Scheme

Transcription

1 Codes and Designs in the Grassmann Scheme Tuvi Etzion Computer Science Department Technion -Israel Institute of Technology ALGEBRAIC COMBINATORICS AND APPLICATIONS ALCOMA10, Thurnau, Germany, 13 April 2010 Joint work with Alexander Vardy

2 Outline Background Bounds on the Sizes of Codes Known Constructions for Constant Dimension Codes New Construction for Constant Weight Codes s for q-covering Designs Bounds on the Size of q-covering Designs

3 Background The projective space of order n over the finite field F q, denoted as P q (n), is the set of all subspaces of the vector space F n q. The natural measure of distance in P q (n) is given by d(u, V ) def = dim U + dim V 2 dim ( U V ) for all U, V P q (n). C P q (n) is an (n, M, d) code in projective space if C = M and d(u, V ) d for all U, V in C. Koetter and Kschischang [2007] showed that codes in P q (n) are precisely what is needed for error-correction in networks.

4 Background Given an integer 0 k n, the set of all subspaces of F n q with dimension k is known as a Grassmannian, and denoted by G q (n, k). C G q (n, k) is an (n, M, d, k) code in the Grassmannian if C = M and d(u, V ) d for all U, V in C. A q-analog t (n, k, λ) design is a set S of k-dimensional subspaces (called blocks) from F n q, such that each t-dimensional subspace of F n q is a subspace of exactly λ blocks from S.

5 Background q-analog designs: Thomas [1987, 1996] Suzuki [1990, 1992] Miyakawa, Munemasa, and Yosihiara [1995] Itoh [1998] Ahlswede, Aydinian, and Khachatrian [2001] Schwartz and Etzion [2002] Braun, Kerber, and Laue [2005]

6 Bounds on the Sizes of Codes A Steiner structure S q [r, k, n] is a collection S of elements from G q (n, k) such that each element from G q (n, r) is contained in exactly one element of S. Let A q (n, d, k) denote the maximum number of codewords in an (n, M, d, k) code in G q (n, k). A q (n, 2δ + 2, k) n k δ k k δ q Steiner structure S q [k δ, k, n] exists. q with equality holds if and only if a

7 Bounds on the Sizes of Codes For a set S G q (n, k) let S be the orthogonal complement of S: S = {A : A S}, where A G q (n, n k) is the orthogonal complement of the subspace A. (complements) A q (n, d, k) = A q (n, d, n k).

8 Bounds on the Sizes of Codes (Johnson) A q (n, 2δ, k) qn 1 q k 1 A q(n 1, 2δ, k 1). A q (n, 2δ, k) qn 1 q n k 1 A q(n 1, 2δ, k). Corollary A q (n, 2δ, k) qn 1 q k 1 qn 1 1 q k 1 1 qn+1 r 1 q k+1 r 1. [ [ n k δ + 1 k k δ + 1 ] q ] q Lemma A q (n, 2k, k) q n 1 q k 1 if n 0 (mod k) 1

9 Constant Dimension Codes (Steiner structure) Construction Let n = sk, r = qn 1 q k 1, and let α be a primitive element in GF(qn ). For each i, 0 i r 1, we define H i = {α i,α r+i,α 2r+i,...,α (qk 2)r+i }. The set {H i : 0 i r 1} is a Steiner structure S q [1, k, n]. Let n r (mod k). Then, for all q, we have A q (n, 2k, k) qn q k (q r 1) 1 q k 1

10 Constant Dimension Codes (Lifted Codes) We can represent X G q (n, k) by the k linearly independent vectors from X which form a unique k n generator matrix in reduced row echelon form, denoted by RE(X ), and defined by: The leading coefficient of a row is always to the right of the leading coefficient of the previous row. All leading coefficients are ones. Every leading coefficient is the only nonzero entry in its column. For each X G q (n, k) we associate a binary vector of length n and weight k, v(x ), called the identifying vector of X, where the ones in v(x ) are in the positions where RE(X ) has the leading ones.

11 Constant Dimension Codes (Lifted Codes) Example Let X be the subspace in G 2 (7, 3) with the following generator matrix in reduced row echelon form: RE(X ) = Its identifying vector is v(x ) = , and its echelon Ferrers form, Ferrers diagram, and Ferrers tableaux form are given by ,, and

12 Constant Dimension Codes (Lifted Codes) For two k η matrices A and B over F q the rank distance is defined by d R (A, B) def = rank(a B). A code C is an [m η, ρ, δ] rank-metric code if its codewords are k η matrices over F q, they form a linear subspace of dimension ρ of F k η q, and for A, B C we have that d R (A, B) δ. Let C be an [k η, ρ, δ] rank-metric code. The subspaces spanned by the set of matrices in reduced row echelon form {[I k A] : A C} form a (k + η, q ρ, 2δ, k) code (identifying vector ).

13 Constant Dimension Codes (Lifted Codes) First codes constructed: Koetter and Kschischang [2007]. First lifted codes: Silva, Koetter and Kschischang [2008] Multilevel Construction: Etzion and Silberstein [2009]

14 Constant Dimension Codes (Cyclic Codes) (Cyclic codes) Let α be a primitive element of GF(2 n ). We say that a code C is cyclic if it has the following property: {0,α i 1,α i 2,...,α i m} is a codeword of C, so is its cyclic shift {0,α i 1+1,α i 2+1,...,α i m+1 }. In other words, if we map each vector space V C into the corresponding binary characteristic vector x V = (x 0, x 1,..., x 2n 2) given by x i = 1 if α i V and x i = 0 if α i V then the set of all such characteristic vectors is closed under cyclic shifts.

15 Constant Dimension Codes (cyclic Codes) A 2 (8, 4, 3) 1275 (compare to A 2 (8, 4, 3) 1493) A 2 (9, 4, 3) 5694 (compare to A 2 (9, 4, 3) 6205) A 2 (10, 4, 3) (compare to A 2 (10, 4, 3) 24698) A 2 (11, 4, 3) (compare to A 2 (11, 4, 3) 99718) A 2 (12, 4, 3) (compare to A 2 (12, 4, 3) ) A 2 (13, 4, 3) (compare to A 2 (13, 4, 3) ) A 2 (14, 4, 3) (compare to A 2 (14, 4, 3) ) Kohnert and Kurz [2008] Etzion and Vardy [2008]

16 New Construction for Constant Weight Codes Construction (COS - cosets of constant dimension code) Let C be an (n, M, d, k) code. From X = {0,α 1,...,α 2k 1} C we form the following set of words with weight 2 k : C X = {{β, β + α 1, β + α 2,..., β + α 2k 1} : β F n 2}. The words of C x represent the cosets of the the k-dimensional subspace X. Therefore, C X = 2 n k. We define our code C as the union of all the sets C X over all codewords of C, i.e., C = X C C X = {{β, β + α 1, β + α 2,..., β + α 2k 1} : {0,α 1,...,α 2k 1} C, β F n 2}.

17 New Construction for Constant Weight Codes If C is an [n, M, d = 2t, k] code then C of Construction COS is a (2 n, 2 n k M, 2 k+1 2 k t+1, 2 k ) code. Example Let C be and an [n, (2n 1)(2 n 1 1) 3, 2, 2] code which consists of all 2-dimensional subspaces of F n 2. C is a (2n, (2n 1)(2 n 1 1)2 n 2 3, 4, 4) code forming the codewords of weight four in the extended Hamming code of length 2 n, i.e., a Steiner system S(3, 4, 2 n ). Example Let C be an [n, 2 n 1, 2, n 1] code (all (n 1)-dimensional subspaces of F n 2 ). C is a (2n, 2 n+1 2, 2 n 1, 2 n 1 ). If we join to C the allone and the allzero codewords then the formed code is a Hadamard code (a Hadamard matrix and its complement).

18 New Construction for Constant Weight Codes A(n, d, w) is the maximum size of a binary constant weight code of length n, weight w, and minimum Hamming distance d. (Johnson) If n w > 0 then n A(n, d, w) w A(n 1, d, w 1).

19 New Construction for Constant Weight Codes (Agrell, Vardy, Zeger) If b > 0 then where b = δ A(n, 2δ, w) w(n w) n δ, b + n { M 2 M w } { M n w } n n M = A(n, 2δ, w). {x} = x x

20 New Construction for Constant Weight Codes A(2 2m 1 1, 2 m+1 4, 2 m 1) = 2 m + 1. A(2 2m 1, 2 m+1 4, 2 m ) = 2 2m m 1. Proof. The upper bound A(2 2m 1 1, 2 m+1 4, 2 m 1) 2 m + 1 is a direct application of AVZ. Using this bound in Johnson we obtain A(2 2m 1, 2 m+1 4, 2 m ) 2 2m m 1. By applying Construction COS on a [2m 1, 2 m + 1, 2m 2, m] code we obtain a (2 2m 1, 2 2m m 1, 2 m+1 4, 2 m ) code. Hence, A(2 2m 1, 2 m+1 4, 2 m ) 2 2m m 1 and thus A(2 2m 1, 2 m+1 4, 2 m ) = 2 2m m 1. By shortening the (2 2m 1, 2 2m m 1, 2 m+1 4, 2 m ) code we obtain a (2 2m 1 1, 2 m + 1, 2 m+1 4, 2 m 1) code and hence A(2 2m 1 1, 2 m+1 4, 2 m 1) = 2 m + 1.

21 New Construction for Constant Weight Codes If a Steiner Structure S 2 [2, k, n] exists then a Steiner system S(3, 2 k, 2 n ) exists. If a Steiner Structure S 2 [2, 3, 7] exists then a Steiner system S(3, 8, 128) exists.

22 s for q-covering Designs A q-covering design C q [n, k, r] is a collection S of elements from G q (n, k) such that each element of G q (n, r) is contained in at least one element of S. A q-turán design T q [n, k, r] is a collection S of elements from G q (n, r) such that each element of G q (n, k) contains at least one element from S.

23 s for q-covering designs The q-covering number C q (n, k, r) is the minimum size of a q-covering design C q [n, k, r]. The q-turán number T q (n, k, r) is the minimum size of a q-turán design T q [n, k, r].

24 Basic Bounds on q-covering numbers S is a q-covering design C q [n, k, r] if and only if S is a q-turán design T q [n, n r, n k]. Corollary C q (n, k, r) = T q (n, n r, n k).

25 Basic Bounds on q-covering numbers n r C q (n, k, r) k r q with equality holds if and only if a Steiner structure S q [r, k, n] exists. q [ n k + r T q (n, k, r) r ]. q Corollary [ n k + r C q (n, k, r) = T q (n, n r, n k) r ]. q

26 Optimal q-covering Designs C q (n, k, 1) = T q (n, n 1, n k) = Sq [1, k, n] = q n 1 q k 1, whenever k divides n. If 1 k n, then C q (n, k, 1) = qn 1 q k 1. If 1 k n 1, then C q (n, n 1, k) = qk+1 1 q 1.

27 Upper Bounds on the Size of q-covering Designs (Recursive Construction) C q (n, k, r) q n k C q (n 1, k 1, r 1) + C q (n 1, k, r). Proof. We represent F n q by {(α, β) : α F n 1 q, β F q }. Let S 1 be a q-covering design C q [n 1, k 1, r 1] and S 2 be a q-covering design C q [n 1, k, r]. We form a set S as follows: For each subspace P = {0,α 1,,α q k 1 1} S 1 let P 1 = P, P 2,..., P q n k be the disjoint cosets of P in F n 1 q. Let β 0 = 0, β 1,..., β q n k be any q n k coset representatives, i.e., β i P i, 1 i q n k. For each 1 i q n k we form the subspace {(α 1, 0),, (α q k 1 1, 0), (β i, 1)} in S. For each subspace {0,α 1,,α qk 1} S 2 the subspace {(0, 0), (α 1, 0),, (α qk 1, 0)} is formed in S. S is a q-covering design C q [n, k, r] and the theorem follows.

28 Lower Bounds on the Size of q-covering Designs C q (n, k, r) qn 1 q k 1 C q(n 1, k 1, r 1). Corollary ] C q (n, k, r) qn 1 q k 1 qn 1 1 q k 1 1 qn+1 r 1 q k+1 r 1 [ n r [ k r q ]. q [ T q (n, r + 1, r) (qn r 1)(q 1) (q r 1) 2 n r 1 ]. q Corollary [ C q (n, k, k 1) (qk 1)(q 1) (q n k 1) 2 n k + 1 ]. q

29 Some Specific Bounds C 2 (5, 3, 2) = 27. (compared to C 2 (5, 3, 2) 23 by previous theorem). 381 C 2 (7, 3, 2) A 2 (7, 4, 3) 381.

30 THANK YOU