Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

Transcription

1 Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound 5th ICMCTA Vihula Manor, August 2017 Daniel Heinlein University of Bayreuth, Germany joined work with Sascha Kurz / 24

2 Notation Grassmannian: Set of k dimensional subspaces in F v q =... of size [ k v ] q = k 1 i=0 ] bijection: τ : [ F v q k q v q i q k q i constant dimension code (cdc): C [ F v q k {X U Fq k v rk(x U ) = k, X U is in rref} [ F v q k subspace distance: d S (U, W ) = dim(u + W ) dim(u W ) maximum cdc size: A q (v, d; k) standard parameters: 2 d/2 k v/2 and 2 q prime power ] ] dominance: means dominates for the parameters applicable q-pochhammer symbol: (a; q) n := n 1 ( i=0 1 aq i ) 2 / 24

3 Upper bounds Anticode type bounds Theorem ((Sphere-packing bound)[kk08, Theorem 6]) A q (v, d; k) (d/2 1)/2 i=0 [ v k ] q q i 2 [ k i ]q [ v k ] i q Theorem ((Anticode bound), cf. [FW86, Theorem 1], [AA09, Lemma 1], [WXSN03, Theorem 5.2]) A q (v, d; k) [ v k ] q [ max{k,v k}+d/2 1 d/2 1 ] q Anticode Sphere Packing 3 / 24

4 Upper bounds Johnson type bounds Theorem ((Johnson type bound I) [XF09, Theorem 2]) If ( q k 1 ) 2 > (q v 1) ( q k d/2 1 ), then A q (v, d; k) ( q k q k d/2) (q v 1) (q k 1) 2 (q v 1) ( q k d/2 1 ). Proposition ((Trivial spread bound) [HK17a, Proposition 1]) For 0 k < v, the bound in Johnson I is applicable iff d = 2 min{k, v k} and k 1. Then, it is equivalent to A q (v, d; k) Trivial spread bound Johnson I q v 1 q min{k,v k} 1. 4 / 24

5 Upper bounds Johnson type bounds, cont. Theorem ((Johnson type bound II) [XF09, Theorem 3], [EV11, Theorem 4,5]) A q (v, d; k) qv 1 q k 1 A q(v 1, d; k 1) A q (v, d; k) qv 1 q v k 1 A q(v 1, d; k) equivalent using orthogonality, cf. [EV11, Section III, esp. Lemma 13] and [HK17a, Proposition 2]. Proposition ([HK17a, Proposition 3]) For k v/2 we have Johnson II.1 Johnson II.2. 5 / 24

6 Upper bounds Johnson type bounds, cont. Corollary ((Recursive Johnson II.1)) A q (v, d; k) q v 1 q v 1 1 q v k+d/2+1 1 q k 1 q k q d/2+1 1 A q(v k +d/2, d; d/2)... Recursive Johnson II.1 Anticode 6 / 24

7 Upper bounds Improved Johnson bound Theorem ((Improved Johnson bound)[kk17, Theorem 3]) Let m = [ v 1 ] q A q (v 1, d; k 1) [ k 1 ]q [ v 1 ] q A q(v 1,d;k 1) [ k1 ] q + [ k 1 ]q δ for some δ N 0. If no q k 1 -divisible multiset of points in F v q of cardinality m exists, then [ v 1 ] q A q (v 1, d; k 1) A q (v, d; k) [ ] k1 δ 1. Theorem ([KK17, Theorem 4]) There exists a q r -divisible multiset of points of cardinality n if and only if there are non-negative integers a 0,..., a r with n = r i=0 a is q i,r and sq i,r = qr i qi+1 1 q 1. Improved Johnson Bound Johnson II.1 q 7 / 24

8 Upper bounds Theorem ((Ahlswede Aydinian) [AA09, Theorem 3]) For integers 0 t < r k, k t m v, and t v m we have A q (v, 2r; k) [ k v ] q A q (m, 2r 2t; k t) t i=0 qi(m+i k) [ k i m [ ] v m. q i ]q Theorem ((Singleton bound) [KK08, Theorem 9]) A q (v, d; k) Ahlswede Aydinian Johnson II.1 Anticode Singleton [ ] v d/2+1 max{k,v k} q 8 / 24

9 Linear Programming Bound Theorem ((Linear Programming Bound) [ZJX11, Section III]) For integers 0 k v and 2 d 2 min{k, v k} such that d is even, we have k A q (v, d; k) max 1 + x i st k i=d/2 i=d/2 Q j (i)x i u j j = 1, 2,..., k and x i 0 i = d/2, d/2 + 1,..., k with E i (j) = i m=0 ( 1)i m q (i m v i = q i [ 2 ki ] q [ v k i 2 )+jm [ ] [ k m k j k i q m ]q, u j = [ v j ]q [ v j 1 Linear Programming Bound Anticode ] q [ ] v k j+m m ]q, Q j(i) = u j v i E i (j) q 9 / 24

10 Upper bounds Theorem ([HKK15, Theorem 1]) A 2 (6, 4; 3) = 77 Proposition ([HHK + ], cf. [HK17b], respective later today) A 2 (8, 6; 4) = / 24

11 Upper bounds MRD Bound Theorem ([ES13, Theorem 10 and 11]) ] Let C be a constant dimension code, with v 2k and [ F v q k minimum subspace distance d, that contains a lifted MRD code. If d = 2(k 1) and k 3, then #C q 2(v k) + A q (v k, 2(k 2); k 1); if d = k, where k is even, then ] #C q (v k)(k/2+1) + [ v k k/2 q q v q v k q k q k/2 + A q (v k, k; k). 11 / 24

12 Upper bounds Partial spreads Theorem ((Spread) [Seg64, VI]) F v q contains a spread if and only if k is a divisor of v. Theorem ((Beutelspacher) [Beu75]) For positive integers v, k satisfying v = tk + r, t 2 and 1 r k 1 we have A q (v, 2k; k) 1 + t 1 i=1 qik+r = qv q k+r +q k 1 with equality for q k 1 r = / 24

13 Upper bounds Partial spreads, cont. Theorem ((Drake Freeman) [DF79, Corollary 8]) If v = kt + r with 0 < r < k, then t 1 A q (v, 2k; k) q ik+r θ 1 = q r qkt 1 q k 1 i=0 where 2θ = 1 + 4q k (q k q r ) (2q k 2q r + 1). Theorem ((Kurz 2.10) [Kur17, Theorem 2.10]) θ 1, For integers r 1, t 2, y max{r, 2}, z 0 with λ = q y, y k, k = [ 1 r ] q + 1 z > r, v = kt + r, and l = qv k q r, we have q k 1 A q (v, 2k; k) lq k + λ λ (λ (z + y 1)(q 1) 1). Kurz 2.10 Drake Freeman 13 / 24

14 Upper bounds Partial spreads, cont. Theorem ((Năstase Sissokho) [NS16, Theorem 5]) Suppose v = tk + r with t 1 and 0 < r < k. If k > [ r 1 ] q then A q (v, 2k; k) = 1 + t 1 i=1 qik+r = qv q k+r +q k 1 q k 1. Theorem ((Kurz 2.9)[Kur17, Theorem 2.9]) For integers r 1, t 2, u 0, and 0 z [ r 1 ] q /2 with k = [ r 1 ] q + 1 z + u > r we have A q (v, 2k; k) lq k z(q 1), where l = qv k q r q k 1 and v = kt + r special cases Kurz 2.10 Drake Freeman Kurz 2.9 Năstase Sissokho Beutelspacher 14 / 24

15 Overview upper bounds for cdcs for 2 d/2 k v/2 and 2 q prime power Ahlswede Aydinian Johnson II.1 Recursive Johnson II.1 Anticode Sphere Packing Improved Johnson bound Johnson II.1 Johnson II.2 Linear Programming Bound Anticode Singleton 2 special cases Spread Kurz 2.10 Drake Freeman Trivial spread bound Johnson I Kurz 2.9 Năstase Sissokho Beutelspacher 21 special cases 15 / 24

16 Revisited linkage construction Theorem ([GLT16, Theorem 2.3], cf. [ST15, Corollary 39]) Let C i be a (v i, N i, d i ; k) q constant dimension code for i = 1, 2 and let C r be a (k v 2, N r, d r ) q linear rank metric code. Then {τ 1 (τ(u) M) : U C 1, M C r } {τ 1 (0 k v1 τ(w )) : W C 2 } is a (v 1 + v 2, N 1 N R + N 2, min{d 1, d 2, 2d r }; k) q constant dimension code. Theorem ([HK17a, Theorem 19]) Let C i be a (v i, N i, d i ; k) q constant dimension code for i = 1, 2, d 2N 0 and let C r be a (k (v 2 k + d/2), N r, d r ) q linear rank metric code. Then {τ 1 (τ(u) M) : U C 1, M C r } {τ 1 (0 k m τ(w )) : W C 2 } is a (v 1 + v 2 k + d/2, N 1 N R + N 2, min{d 1, d 2, 2d r, d}; k) q constant dimension code with m = v 1 k + d/2. 16 / 24

17 Asymptotic bounds Proposition ([HK17a, Proposition 7]) For k v k the ratio of the size of a lifted MRD code divided by the size of the Singleton bound converges for v monotonically decreasing to (1/q; 1/q) k d/2+1 (1/2; 1/2) > Proposition ([HK17a, Proposition 8], cf. [ES13, Lemma 9]) For k v k the ratio of the size of a lifted MRD code divided by the size of the Anticode bound converges for v monotonically decreasing to (1/q;1/q) k (1/q;1/q) q d/2 1 q 1 (1/q; 1/q) k 2 (1/2; 1/2) > Improved Johnson bound (and especially Johnson II.1) does not improve this limit. 17 / 24

18 Codes better than the MRD bound Proposition ([HK17a, Proposition 10]) For q 3 we have lim v A q(v,4;3) [ ] q 2v 6 + v 3 2 q Proposition ([HK17a, Proposition 11]) For v 19 we have A 2 (v,4;3) [ ] 2 2v 6 + v q / 24

19 Thank you for your attention I R Ahlswede and H Aydinian. On error control codes for random network coding. In Network Coding, Theory, and Applications, NetCod 09. Workshop on, pages IEEE, A. Beutelspacher. Partial spreads in finite projective spaces and partial designs. Mathematische Zeitschrift, 145(3): , D.A. Drake and J.W. Freeman. Partial t-spreads and group constructible (s, r, µ)-nets. Journal of Geometry, 13(2): , Tuvi Etzion and Natalia Silberstein. Codes and designs related to lifted MRD codes. IEEE Transactions on Information Theory, 59(2): , / 24

20 Thank you for your attention II Tuvi Etzion and Alexander Vardy. Error-correcting codes in projective space. IEEE Transactions on Information Theory, 57(2): , P. Frankl and R. M. Wilson. The Erdős-Ko-Rado theorem for vector spaces. Journal of Combinatorial Theory, Series A, 43(2): , Heide Gluesing-Luerssen and Carolyn Troha. Construction of subspace codes through linkage. Advances in Mathematics of Communications, 10(3): , Daniel Heinlein, Thomas Honold, Michael Kiermaier, Sascha Kurz, and Alfred Wassermann. Classification of optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6. in preparation. 20 / 24

21 Thank you for your attention III Daniel Heinlein and Sascha Kurz. Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound. arxiv preprint arxiv: , Daniel Heinlein and Sascha Kurz. A new upper bound for subspace codes. arxiv preprint , Thomas Honold, Michael Kiermaier, and Sascha Kurz. Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4. In Topics in finite fields, volume 632 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, / 24

22 Thank you for your attention IV Ralf Kötter and Frank R. Kschischang. Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory, 54(8): , Michael Kiermaier and Sascha Kurz. An improvement of the johnson bound for subspace codes. arxiv preprint arxiv: , Sascha Kurz. Packing vector spaces into vector spaces. The Australasian Journal of Combinatorics, 68(1): , Esmeralda Năstase and Papa Sissokho. The maximum size of a partial spread in a finite projective space. arxiv preprint , / 24

23 Thank you for your attention V B. Segre. Teoria di galois, fibrazioni proiettive e geometrie non desarguesiane. Annali di Matematica Pura ed Applicata, 64(1):1 76, Natalia Silberstein and Anna-Lena Trautmann. Subspace codes based on graph matchings, ferrers diagrams, and pending blocks. IEEE Transactions on Information Theory, 61(7): , Huaxiong Wang, Chaoping Xing, and Reihaneh Safavi-Naini. Linear authentication codes: bounds and constructions. IEEE Transactions on Information Theory, 49(4): , Shu-Tao Xia and Fang-Wei Fu. Johnson type bounds on constant dimension codes. Designs, Codes and Cryptography, 50(2): , / 24

24 Thank you for your attention VI Zong-Ying Zhang, Yong Jiang, and Shu-Tao Xia. On the linear programming bounds for constant dimension codes. In Network Coding (NetCod), 2011 International Symposium on, pages 1 4. IEEE, / 24