Maximum Rank Distance Codes are Generic Gabidulin Codes are Not

Transcription

1 Maximum Rank Distance Codes are Generic Gabidulin Codes are Not Anna-Lena Horlemann-Trautmann Algorithmics Laboratory, EPF Lausanne, Switzerland April 4th, 2016 Network Coding and Designs, Dubrovnik joint work with Alessandro Neri, Tovohery Randrianarisoa, Joachim Rosenthal

2 Introduction Codes achieving the Singleton bound for rank metric are called MRD (maximum rank distance) codes. Known since 1978 (Delsarte)/1985 (Gabidulin): General construction for MRD codes for any set of parameters. = Gabidulin codes 1 / 23

3 Introduction Codes achieving the Singleton bound for rank metric are called MRD (maximum rank distance) codes. Known since 1978 (Delsarte)/1985 (Gabidulin): General construction for MRD codes for any set of parameters. = Gabidulin codes Until 2 years ago no really different general construction was known. Now few new results. In the smallest non-trivial case, all MRD codes are Gabidulin. Question: How many (linear) MRD codes are there and how many of those are Gabidulin codes? 1 / 23

4 MRD and Gabidulin Codes 1 MRD and Gabidulin Codes 2 Generic Sets and the Zariski Topology 3 MRD Codes are Generic Sets 4 Non-Gabidulin Codes are Generic Sets 5 Rough Probability Estimation

5 MRD and Gabidulin Codes Rank metric: d R (A, B) := rank(a B), A, B F m n q with ϕ : F n q m Fm n q d R (a, b) := rank(ϕ(a) ϕ(b)), isomorphism. a, b F n q m Definition A linear code C F n q m of dimension k is called an MRD (maximum rank distance) code, if the minimum rank distance of C is equal to n k / 23

6 MRD and Gabidulin Codes Lemma Any MRD code C F n q m G F k n q m in systematic form, i.e. of dimension k has a generator matrix G = [ I k X ] Moreover, all entries in X are from F q m\f q. 3 / 23

7 MRD and Gabidulin Codes Lemma Any MRD code C F n q m G F k n q m in systematic form, i.e. of dimension k has a generator matrix G = [ I k X ] Moreover, all entries in X are from F q m\f q. Question: Which X F k(n k) q m generate what type of code? 3 / 23

8 MRD and Gabidulin Codes MRD Criteria Theorem (Gabidulin) Let G F k n q m be a generator matrix of a rank-metric code C F n qm. Then C is an MRD code if and only if for any A F n k q of rank k, det(ga) 0. 4 / 23

9 MRD and Gabidulin Codes MRD Criteria Let UTn(q) be the subgroup of GL n (q) of upper triangular matrices with all 1 on the diagonal, i.e 1 a 12 a 1n UTn(q) 0 1 a 2n = a ij F q Theorem (HT-Marshall) Let G F k n q m be a generator matrix of a rank-metric code C F n qm. Then C is an MRD code if and only if for any A UTn(q) every maximal minor of GA is non-zero. 5 / 23

10 MRD and Gabidulin Codes Definition Let g 1,..., g n F q m be linearly independent over F q and gcd(s, m) = 1. A code C F n q m with generator matrix G = g 1 g 2... g n g qs 1 g qs 2... gn qs. g qs(k 1) 1 g qs(k 1) 2... gn qs(k 1) is called a generalized Gabidulin code. For s = 1 it is a classical Gabidulin code. For s = 1 it was shown by Delsarte (1978) and Gabidulin (1985) that these codes are MRD. For s > 1 this fact was shown by Kshevetskiy and Gabidulin (2005).. 6 / 23

11 MRD and Gabidulin Codes Theorem (HT-Marshall) An MRD code C F n q m of dimension k is a generalized Gabidulin code if and only if dim(c C (qs) ) = k 1. 7 / 23

12 MRD and Gabidulin Codes Theorem (HT-Marshall) An MRD code C F n q m of dimension k is a generalized Gabidulin code if and only if dim(c C (qs) ) = k 1. If G = [I k X] is generator matrix, then dim(c C (qs) ) = k 1 is equivalent to [ ] Ik X rank I k X (qs ) = k + 1 [ ] Ik X rank 0 X (qs) = k + 1 X rank(x (qs) X) = 1. 7 / 23

13 Generic Sets and the Zariski Topology 1 MRD and Gabidulin Codes 2 Generic Sets and the Zariski Topology 3 MRD Codes are Generic Sets 4 Non-Gabidulin Codes are Generic Sets 5 Rough Probability Estimation

14 Generic Sets and the Zariski Topology A generic property is one that holds almost everywhere. Definition A property of an irreducible algebraic variety is said to be true generically, if it holds on a non-empty Zariski-open subset. 8 / 23

15 Generic Sets and the Zariski Topology A generic property is one that holds almost everywhere. Definition A property of an irreducible algebraic variety is said to be true generically, if it holds on a non-empty Zariski-open subset. Denote by F q the algebraic closure of F q. Definition (Zariski topology) The Zariski topology on F r q can be defined by specifying its closed sets, namely as the algebraic sets: V (S) = {x F r q f(x) = 0, f S}, where S is any set of polynomials in F q [x 1,..., x r ]. 8 / 23

16 Generic Sets and the Zariski Topology The open sets in the Zariski topology on F r q are the complements of a closed set. All sets of the form O = {x F r q f(x) 0, f S} are open since their complement is given by O C = {x F r q f S f(x) = 0}, which is a Zariski-closed set. 9 / 23

17 MRD Codes are Generic Sets 1 MRD and Gabidulin Codes 2 Generic Sets and the Zariski Topology 3 MRD Codes are Generic Sets 4 Non-Gabidulin Codes are Generic Sets 5 Rough Probability Estimation

18 MRD Codes are Generic Sets Recall that C = rowspan[i k X] F n q m is an MRD code if and only if for any A F n k q of rank k, det([i k X]A) 0. The entries of X F k(n k) q m are the variables x 1,..., x k(n k). Since det([i k X]A) F q m[x 1,..., x k(n k) ] we get in the algebraic closure: Lemma The set of non-mrd codes C F n q m Corollary The set of MRD codes C F n q m therefore a generic set. is a Zariski-closed set. is a Zariski-open set and 10 / 23

19 MRD Codes are Generic Sets = For very large field size a random linear code is most likely an MRD code! 11 / 23

20 MRD Codes are Generic Sets = For very large field size a random linear code is most likely an MRD code! MRD is generic! 11 / 23

21 Non-Gabidulin Codes are Generic Sets 1 MRD and Gabidulin Codes 2 Generic Sets and the Zariski Topology 3 MRD Codes are Generic Sets 4 Non-Gabidulin Codes are Generic Sets 5 Rough Probability Estimation

22 Non-Gabidulin Codes are Generic Sets Recall that an MRD code C = rowspan[i k X] F n q m is a generalized Gabidulin code if and only if rank(x X (qs) ) = 1. This condition is equivalent to (if all x i F q ) 2 2 submatrices M of (X X (qs) ) : det(m) = 0. Since det(m) F q m[x 1,..., x k(n k) ] we get in the algebraic closure: 12 / 23

23 Non-Gabidulin Codes are Generic Sets Recall that an MRD code C = rowspan[i k X] F n q m is a generalized Gabidulin code if and only if rank(x X (qs) ) = 1. This condition is equivalent to (if all x i F q ) 2 2 submatrices M of (X X (qs) ) : det(m) = 0. Since det(m) F q m[x 1,..., x k(n k) ] we get in the algebraic closure: Theorem The set of Gabidulin codes C F n q m the set of MRD codes. is a Zariski-closed subset of 12 / 23

24 Non-Gabidulin Codes are Generic Sets = For very large field size a random linear MRD code is most likely not a generalized Gabidulin code! 13 / 23

25 Non-Gabidulin Codes are Generic Sets = For very large field size a random linear MRD code is most likely not a generalized Gabidulin code! Non-Gabidulin is generic for linear MRD codes! 13 / 23

26 Rough Probability Estimation 1 MRD and Gabidulin Codes 2 Generic Sets and the Zariski Topology 3 MRD Codes are Generic Sets 4 Non-Gabidulin Codes are Generic Sets 5 Rough Probability Estimation

27 Rough Probability Estimation Lemma (Schwartz-Zippel) Let f F[x 1, x 2,..., x r ] be a non-zero polynomial of total degree d 0. Let S F and let v 1, v 2,..., v r be selected at random independently and uniformly from S. Then Pr[f(v 1, v 2,..., v r ) = 0] d S. 14 / 23

28 Rough Probability Estimation MRD Codes (with Gabidulin criterion) Consider G = [I k X] and A F n k q of rank k. There are k 1 i=0 (qn q i ) q kn many different A s. Moreover, det(ga) has degree at most k. Hence the product of all these determinants has degree at most kq kn. Theorem The probability that a randomly chosen X F k(n k) q m non-mrd code in F n q m is Pr[P det (x 1, x 2,..., x k(n k) ) = 0] kqkn q m = kqkn m. For m the probability goes to zero! generates a 15 / 23

29 Rough Probability Estimation MRD Codes (with HT-Marshall criterion) Consider G = [I k X]. The product of all maximal minors of G has degree k ( )( ) ( ) k n k n 1 i = (n k). i k i k 1 i=0 On the other hand we have UT n(q) = q n(n 1) 2. Theorem The probability that a randomly chosen X F k(n k) q m non-mrd code in F n q m is generates a Pr[P minor (x 1, x 2,..., x k(n k) ) = 0] q n(n 1) 2 (n k) ( ) n 1 k 1 q m. For m the probability goes to zero! 16 / 23

30 Rough Probability Estimation Gabidulin Codes (s = 1) Recall that rank(x X q ) = 0 cannot generate an MRD code. We have {X X gen. Gabidulin} = {X X gen. MRD} {X rank(x X q ) = 1}. 17 / 23

31 Rough Probability Estimation Gabidulin Codes (s = 1) Recall that rank(x X q ) = 0 cannot generate an MRD code. We have {X X gen. Gabidulin} = Hence {X X gen. MRD} {X rank(x X q ) = 1}. {X X generates Gabidulin code} {X rank(x X q ) 1}, i.e., Pr[X generates Gabidulin code] Pr[ rank(x X q ) 1 ]. }{{} if all 2 2 minors are zero 17 / 23

32 Rough Probability Estimation Gabidulin Codes (s = 1) We will check all non-intersecting 2 2-minors M ij of (X X (q) ), of which we have k n k 2 2 many. Each determinant has degree 2q, hence Pr(M ij = 0) 2q 1 m. Since these determinants are independent we get: Theorem The probability that a randomly chosen X F k(n k) q m Gabidulin code is i,j Pr[M ij = 0] (2q 1 m ) k n k 2 2. generates a For m the probability goes to zero! 18 / 23

33 Rough Probability Estimation For all other s 1 we get the same number, hence for the probability of getting any generalized Gabidulin code we need to multiply with Euler-φ(m) (since s, m are coprime). Theorem The probability that a randomly chosen X F k(n k) q m generalized Gabidulin code is upper bounded by generates a φ(m)(2q 1 m ) k n k 2 2 (m 1)(2q 1 m ) k n k / 23

34 Rough Probability Estimation Upper bound on probabilities of non-mrd n = 4, 5, k = 2, q = 2: Pr Pr m m 20 / 23

35 Rough Probability Estimation Upper bound on probabilities of generalized Gabidulin n = 4, 5, k = 2, q = 2: Pr Pr m m 21 / 23

36 Rough Probability Estimation Upper bound on probabilities that MRD is generalized Gabidulin n = 4, 5, k = 2, q = 2: 1.5e-3 Pr 1.2e-4 Pr 1e-4 1e-3 8e-5 6e-5 5e-4 4e-5 2e m m Pr[Gab MRD] = Pr[MRD Gab] Pr[Gab] Pr[M RD] = Pr[Gab] Pr[M RD] 22 / 23

37 Rough Probability Estimation Conclusions A random linear code is very likely MRD for large field size. A random linear code is very likely non-gabidulin for large field size. Even a random linear MRD code is very likely non-gabidulin for large field size. 23 / 23

38 Rough Probability Estimation Conclusions A random linear code is very likely MRD for large field size. A random linear code is very likely non-gabidulin for large field size. Even a random linear MRD code is very likely non-gabidulin for large field size. Open question: What are all the other MRD codes out there? Thanks for your attention! Questions? Comments? 23 / 23