Phase Transitions of Random Codes and GV-Bounds

Transcription

1 Phase Transitions of Random Codes and GV-Bounds Yun Fan Math Dept, CCNU A joint work with Ling, Liu, Xing Oct 2011 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

2 Phase Transitions of Random Codes and GV-Bounds 1 Phase Transitions 2 Shannon s World 3 Hamming s World 4 Gilbert-Varshamov Bound 5 Phase Transitions of Random Codes 6 Pictures for Phase Transitions of Random Codes Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

3 Phase Transitions: in Physics Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

4 Phase Transitions: in Physics SOLID melting freezing LIQUID vaporization condensation GAS 0 o C 100 o C Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

5 Phase Transitions in Mathematics: Pioneer Random graph: G(n, p) n vertices to each pare of two vertices an edge is settled at probability p Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

6 Phase Transitions in Mathematics: Pioneer Random graph: G(n, p) n vertices to each pare of two vertices an edge is settled at probability p n = 2 a a a a probability 1 p p Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

7 Phase Transitions in Mathematics: Pioneer Random graph: G(n, p) n vertices to each pare of two vertices an edge is settled at probability p n = 2 a a a a probability 1 p p n = 3 a a a a a a a a a a Ja a J probability (1 p) 3 3(1 p) 2 p 3(1 p)p 2 p 3 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

8 Phase Transitions in Mathematics: Pioneer Is G(n, p) connected? Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

9 Phase Transitions in Mathematics: Pioneer Is G(n, p) connected? DISCONNECTED ln n n p increasing p decreasing ln n n CONNECTED Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

10 Phase Transitions in Mathematics: Pioneer lim Pr ( G(n, p) connected ) = n { 0, p < ln n n ; 1, p > ln n n. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

11 Phase Transitions in Mathematics: Pioneer lim Pr ( G(n, p) connected ) = n { 0, p < ln n n ; 1, p > ln n n. published in: P. Erdös, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. vol.5, pp17-61, Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

12 Phase Transitions in Mathematics: Pioneer lim Pr ( G(n, p) connected ) = n { 0, p < ln n n ; 1, p > ln n n. published in: P. Erdös, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. vol.5, pp17-61, usage phase transition in Mathematics appeared first time in: S.Janson, T.Luczak, and A.Rucinski, The Phase Transition Ch.5 in Random Graphs, New York: Wiley, pp , Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

13 Phase Transitions in Mathematics: Linear system Random linear system S over a finite filed F : a 11 x a 1n x n = b 1 S : a m1 x a mn x n = b m Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

14 Phase Transitions in Mathematics: Linear system Random linear system S over a finite filed F : a 11 x a 1n x n = b 1 S : a m1 x a mn x n = b m lim Pr ( S has solutions ) = n { 1, r < 1; 0, r > 1. where r = m n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

15 Phase Transitions in Mathematics: Linear system Random linear system S over a finite filed F : a 11 x a 1n x n = b 1 S : a m1 x a mn x n = b m lim Pr ( S has solutions ) = n { 1, r < 1; 0, r > 1. where r = m n SOLUTION 1 r increasing r decreasing 1 NO SOLUTION Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

16 Phase Transitions in Mathematics: Linear system A proof Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

17 Phase Transitions in Mathematics: Linear system A proof A = (a ij ) m n = ( A 1 A n ), b = (b1,, b m ) T. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

18 Phase Transitions in Mathematics: Linear system A proof A = (a ij ) m n = ( A 1 A n ), b = (b1,, b m ) T. r < 1 Pr ( ranka < m ) W F m,dim W =m 1 q m (1/q) n = q (r 1)n 0. Pr ( all A j contained in W ) lim Pr ( solution ) lim Pr ( ranka = m ) = 1 n n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

19 Phase Transitions in Mathematics: Linear system A proof A = (a ij ) m n = ( A 1 A n ), b = (b1,, b m ) T. r < 1 Pr ( ranka < m ) W F m,dim W =m 1 q m (1/q) n = q (r 1)n 0. Pr ( all A j contained in W ) lim Pr ( solution ) lim Pr ( ranka = m ) = 1 n n r > 1 Pr ( solution ) = Pr ( b Span(A 1,, A n ) ) q n /q m = q (1 r)n 0. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

20 Shannon s World F : alphabet, cardinality F = q. Message word w F k Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

21 Shannon s World F : alphabet, cardinality F = q. Message word w F k encoding w E: F k F n E(w) CHANNEL E(w) + e noise e D: F n F k D ( E(w) + e ) decoding Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

22 Shannon s World F : alphabet, cardinality F = q. Message word w F k encoding w E: F k F n E(w) CHANNEL E(w) + e noise e Does D ( E(w) + e ) = w? D: F n F k D ( E(w) + e ) decoding Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

23 Shannon s World: Binary symmetric channels b b 1 p - Z p ZZ > p Z ZZ~ - 1 p b b p=false probability 1 p=true probability Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

24 Shannon s World: Binary symmetric channels b b 1 p - Z p ZZ > p Z ZZ~ - 1 p b b p=false probability 1 p=true probability Transition probability matrix: ( 1 p p ) p 1 p Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

25 Shannon s World: Binary symmetric channels b b 1 p - Z p ZZ > p Z ZZ~ - 1 p b b p=false probability 1 p=true probability Transition probability matrix: ( 1 p p ) p 1 p Entropy: H(p) = p log 2 p (1 p) log 2 (1 p) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

26 Shannon s World: q-ary symmetric channels b b H 1 p A HHHH * p q 1 HHj H A 1 p - J HHHH p q 1 JJ A H@ AA J. JJ. A AA J. JJ.. AU. b b A J AA JJ^ p = false probability 1 p = true probability transition probability matrix: p p 1 p q 1 q 1 p p q 1 1 p q 1 p p q 1 q 1 1 p Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

27 Shannon s World: q-ary symmetric channels b b H 1 p A HHHH * p q 1 HHj H A 1 p - J HHHH p q 1 JJ A H@ AA J. JJ. A AA J. JJ.. AU. b b A J AA JJ^ p = false probability 1 p = true probability transition probability matrix: p p 1 p q 1 q 1 p p q 1 1 p q 1 p p q 1 q 1 1 p q-ary Entropy: H q (p) = log q (q 1) p log q p (1 p) log q (1 p) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

28 Shannon s World: Shannon s Theorem Coding device = Rate r = k/n E : F k F n + D : F n F k Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

29 Shannon s World: Shannon s Theorem Coding device = E : F k F n + D : F n F k Rate r = k/n Shannon s Theorem If r < 1 H q (p) then there exists a coding device of rate r such that lim Pr(D( E(w) + e) = w ) = 1. n If r > 1 H q (p) then, for any coding device of rate r, lim Pr(D( E(w) + e) = w ) = 0. n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

30 Shannon s World: Shannon s Theorem c = 1 H q (p): the capacity of the q-ary symmetric channel. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

31 Shannon s World: Shannon s Theorem c = 1 H q (p): the capacity of the q-ary symmetric channel. HIGH FIDELITY c r increasing r decreasing c NO FIDELITY Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

32 Shannon s World: Shannon s Theorem C. E. Shannon, A mathematical theory of communication, Bell Sys. Tech. Journal, vol.27, pp , 623C655, If r < c, lim Pr(D( E(w) + e) = w ) = 1. n If r > c, there is a b < 1 such that lim n Pr(D( E(w) + e) = w ) < b. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

33 Shannon s World: Shannon s Theorem C. E. Shannon, A mathematical theory of communication, Bell Sys. Tech. Journal, vol.27, pp , 623C655, If r < c, lim Pr(D( E(w) + e) = w ) = 1. n If r > c, there is a b < 1 such that lim n Pr(D( E(w) + e) = w ) < b. Jacob Wolfowitz, The coding of messages subject to chancs errors, Illinois J. Math., vol.1, pp , If r > c, lim n Pr(D( E(w) + e) = w ) = 0. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

34 Hamming s World Hamming distance: d H (x, x ) = {1 i n x i x i }, x, x F n Codes: C F n rate: R(C) = log q ( C )/n, i.e. C = F R(C)n minimal distance: d H (C) = min c c C d H(c, c ) relative distance: δ H (C) = d H (C)/n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

35 Hamming s World Hamming distance: d H (x, x ) = {1 i n x i x i }, x, x F n Codes: C F n rate: R(C) = log q ( C )/n, i.e. C = F R(C)n minimal distance: d H (C) = min c c C d H(c, c ) relative distance: δ H (C) = d H (C)/n Linear codes: if F is a finite field, and C is a subspace of F n weight: w H (x) = {1 i n x i 0} minimal weight: w H (C) = min 0 c C w H(c) d H (C) = w H (C) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

36 Hamming s World: Good codes Good C: R(C) is large, and δ H (C) is large Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

37 Hamming s World: Good codes Good C: R(C) is large, and δ H (C) is large Trade off between rate and relative distance? Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

38 Hamming s World: Upper bounds Let δ 0 = 1 q 1 Given δ H (C) = δ, 0 < δ < δ 0 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

39 Hamming s World: Upper bounds Let δ 0 = 1 q 1 Given δ H (C) = δ, 0 < δ < δ 0 Many functions bound up r = R(C): Singleton bound: r 1 δ Plotkin bound: r 1 δ δ 0 Hamming bound: r 1 H q ( δ 2 ) Elias bound: r 1 H q (δ 0 δ 0 (δ 0 δ)) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

40 Hamming s World: Lower bounds When we are given δ H (C) = δ, to explore good codes, we are in fact concerned with how large r = R(C) could reach. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

41 Gilbert-Varshamov Bound: Function GV q (δ) GV q (δ) = 1 H q (δ), δ (0, δ 0 ) GV q(δ) 6 1 δ 0 - δ Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

42 Gilbert-Varshamov Bound: GV-bound Asymptotic Gilbert-Varshamov Bound For any r < GV q (δ), there exist codes C over F (of large enough length) with rate r and relative distance > δ. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

43 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

44 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Take c 1 F n ; take c 2 F n B(c 1, d); Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

45 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Take c 1 F n ; take c 2 F n B(c 1, d); once F n m B(c i, d), take c m+1 F n m B(c i, d); i=1 i=1 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

46 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Take c 1 F n ; take c 2 F n B(c 1, d); once F n m B(c i, d), take c m+1 F n m B(c i, d); i=1 up to M B(c i, d) = F n. i=1 i=1 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

47 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Take c 1 F n ; take c 2 F n B(c 1, d); once F n m B(c i, d), take c m+1 F n m B(c i, d); i=1 up to M B(c i, d) = F n. i=1 M V q (n, d) = M i=1 V q(n, d) i=1 M B(c i, d) = qn i=1 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

48 GV-bound: Gilbert s argument, greedy algorithm d = δn, k = rn ; M = q k B(x, d) =the Hamming ball, V q (n, d) = B(x, d) Take c 1 F n ; take c 2 F n B(c 1, d); once F n m B(c i, d), take c m+1 F n m B(c i, d); i=1 up to M B(c i, d) = F n. i=1 M V q (n, d) = M i=1 V q(n, d) i=1 M B(c i, d) = qn r = log q M/n 1 log q V q (n, d)/n 1 H q (δ) = GV q (δ) i=1 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

49 Varshamov s argument, probabilistic method Varshamov Select a linear code L of rate r < GV q (δ) uniformly at random from F n where F is a finite field, then lim Pr(δ H(L) > δ) = 1 n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

50 Varshamov s argument, probabilistic method Varshamov Select a linear code L of rate r < GV q (δ) uniformly at random from F n where F is a finite field, then lim Pr(δ H(L) > δ) = 1 n Random linear map: g : F k F n, or, random matrix G = (g ij ) m n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

51 Varshamov s argument, probabilistic method Varshamov Select a linear code L of rate r < GV q (δ) uniformly at random from F n where F is a finite field, then lim Pr(δ H(L) > δ) = 1 n Random linear map: g : F k F n, or, random matrix G = (g ij ) m n For any 0 b F k, Pr ( w H (g(b)) d ) = V q (n, d)/q n q GVq(δ)n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

52 Varshamov s argument, probabilistic method Varshamov Select a linear code L of rate r < GV q (δ) uniformly at random from F n where F is a finite field, then lim Pr(δ H(L) > δ) = 1 n Random linear map: g : F k F n, or, random matrix G = (g ij ) m n For any 0 b F k, Pr ( w H (g(b)) d ) = V q (n, d)/q n q GVq(δ)n Pr(δ H (C) δ) = Pr (w H (g(b)) d) 0 b F k Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

53 Varshamov s argument, probabilistic method Varshamov Select a linear code L of rate r < GV q (δ) uniformly at random from F n where F is a finite field, then lim Pr(δ H(L) > δ) = 1 n Random linear map: g : F k F n, or, random matrix G = (g ij ) m n For any 0 b F k, Pr ( w H (g(b)) d ) = V q (n, d)/q n q GVq(δ)n Pr(δ H (C) δ) = Pr (w H (g(b)) d) 0 b F k Pr(δ H (C) δ) q k V q (n, d)/q n q rn GVq(δ)n 0 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

54 GV-Bound: Comparison with Shannon s world Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

55 GV-Bound: Comparison with Shannon s world q-ary GV-bound coincides exactly with the Shannon s capacity of q-ary symmetric channels Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

56 GV-Bound: Comparison with Shannon s world q-ary GV-bound coincides exactly with the Shannon s capacity of q-ary symmetric channels Varshamov s argument deals with random objects by means of probabilistic methods Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

57 GV-Bound: Comparison with Shannon s world q-ary GV-bound coincides exactly with the Shannon s capacity of q-ary symmetric channels Varshamov s argument deals with random objects by means of probabilistic methods What happen if r is beyond GV-bound? Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

58 GV-Bound: Beyond GV-bound Codes of relative distance > δ and rate r beyond GV-bound? Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

59 GV-Bound: Beyond GV-bound Codes of relative distance > δ and rate r beyond GV-bound? Famous work: Yes if q 49, algebraic geometry codes M. A. Tsfasman, S.G. Vladuts, T. Zink, Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachrichten, vol.104, pp.13-28, Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

60 GV-Bound: Beyond GV-bound Codes of relative distance > δ and rate r beyond GV-bound? Famous work: Yes if q 49, algebraic geometry codes M. A. Tsfasman, S.G. Vladuts, T. Zink, Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachrichten, vol.104, pp.13-28, For q = 2, no code C with δ H (C) > δ and rate R(C) > GV q (δ) is reported Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

61 GV-Bound: Beyond GV-bound Codes of relative distance > δ and rate r beyond GV-bound? Famous work: Yes if q 49, algebraic geometry codes M. A. Tsfasman, S.G. Vladuts, T. Zink, Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachrichten, vol.104, pp.13-28, For q = 2, no code C with δ H (C) > δ and rate R(C) > GV q (δ) is reported A guess: no code of rel. dist. > δ and rate > GV q (δ) for q = 2 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

62 GV-Bound: Arbitrary random codes F : an alphabet C: random code of rate r of F n (selected uniformly at random from F n ) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

63 GV-Bound: Arbitrary random codes F : an alphabet C: random code of rate r of F n (selected uniformly at random from F n ) lim Pr(δ H(C) > δ) = 1, n if r < 1 2 GV q(δ) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

64 GV-Bound: Arbitrary random codes F : an alphabet C: random code of rate r of F n (selected uniformly at random from F n ) lim Pr(δ H(C) > δ) = 1, n if r < 1 2 GV q(δ) It follows from: Alexander Barg, G. David Forney, Random codes: Minimum distances and error exponents, IEEE Trans. Inform. Theory, vol.48, pp , Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

65 Phase Transitions of Random Codes: What we do?? F : finite field L: random linear code of rate r of F n lim Pr(δ H(L) > δ) = n { 1, if r < GVq (δ) ;??, if r > GV q (δ). Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

66 Phase Transitions of Random Codes: What we do?? F : finite field L: random linear code of rate r of F n lim Pr(δ H(L) > δ) = n F : alphabet C: random code of rate r of F n lim Pr(δ H(C) > δ) = n { 1, if r < GVq (δ) ;??, if r > GV q (δ). { 1, if r < 1 2 GV q(δ) ;??, if r > 1 2 GV q(δ). Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

67 Phase Transitions of Random Codes: Linear case F : finite field L: random linear code of rate r of F n Our Result lim Pr(δ H(L) > δ) = n { 1, if r < GVq (δ) ; 0, if r > GV q (δ). Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

68 Phase Transitions of Random Codes: Linear case F : finite field L: random linear code of rate r of F n Our Result lim Pr(δ H(L) > δ) = n { 1, if r < GVq (δ) ; 0, if r > GV q (δ). Is δ H (L) > δ? YES GV q (δ) r increasing r decreasing GV q (δ) NO Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

69 Linea case: Sketch of proof Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

70 Linea case: Sketch of proof Random linear map g : F k F n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

71 Linea case: Sketch of proof Random linear map Random variables X b = g : F k F n { 1, w H ( g(b) ) δn; 0, otherwise. b F k. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

72 Linea case: Sketch of proof Random linear map Random variables X b = g : F k F n { 1, w H ( g(b) ) δn; 0, otherwise. b F k. Random variable X = 0 b F k X b. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

73 Linea case: Sketch of proof Random linear map Random variables X b = g : F k F n { 1, w H ( g(b) ) δn; 0, otherwise. b F k. Random variable X = 0 b F k X b. Expectation E(X b ) = V q (n, δn)/q n = V q (n, δn)/q n q GVq(δ)n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

74 Linea case: Sketch of proof Random linear map Random variables X b = g : F k F n { 1, w H ( g(b) ) δn; 0, otherwise. b F k. Random variable X = 0 b F k X b. Expectation E(X b ) = V q (n, δn)/q n = V q (n, δn)/q n q GVq(δ)n Expectation E(X ) = (q k 1)E(X b ) { 0, r < GV q (δ);, r > GV q (δ). Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

75 Linea case: Sketch of proof: Case r < GV q (δ) Case r < GV q (δ) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

76 Linea case: Sketch of proof: Case r < GV q (δ) Case r < GV q (δ) By Markov s inequality, lim Pr(X 1) lim E(X ) = 0 n n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

77 Linea case: Sketch of proof: Case r < GV q (δ) Case r < GV q (δ) By Markov s inequality, lim Pr(X 1) lim E(X ) = 0 n n so lim Pr ( δ H (L) > δ ) = lim Pr(X = 0) = 1 n n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

78 Linea case: Sketch of proof: Case r > GV q (δ) Case r > GV q (δ) Second moment method: Pr(X 1) E(X b ) E(X X b = 1) 0 b F k Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

79 Linea case: Sketch of proof: Case r > GV q (δ) Case r > GV q (δ) Second moment method: Pr(X 1) E(X b ) E(X X b = 1) 0 b F k we can compute that E(X X b = 1) = (q k q)e(x b ) + (q 1) Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

80 Linea case: Sketch of proof: Case r > GV q (δ) Case r > GV q (δ) Second moment method: we can compute that Pr(X 1) E(X b ) E(X X b = 1) 0 b F k E(X X b = 1) = (q k q)e(x b ) + (q 1) from that lim n qk E(X b ) = lim E(X ) =, we have n q k E(X b ) lim Pr(X 1) lim n n (q k q)e(x b ) + (q 1) = 1 Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

81 Linea case: Sketch of proof: Case r > GV q (δ) Case r > GV q (δ) Second moment method: we can compute that Pr(X 1) E(X b ) E(X X b = 1) 0 b F k E(X X b = 1) = (q k q)e(x b ) + (q 1) from that lim n qk E(X b ) = lim E(X ) =, we have n so q k E(X b ) lim Pr(X 1) lim n n (q k q)e(x b ) + (q 1) = 1 lim Pr ( δ H (L) > δ ) = lim Pr(X = 0) = 0 n n Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

82 Phase transitions of Random Codes: Arbitrary case F : an alphabet C: random code of rate r of F n Our Result lim Pr(δ H(C) > δ) = n 1, if r < 1 2 GV q(δ) ; 0, if r > 1 2 GV q(δ). Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

83 Phase transitions of Random Codes: Arbitrary case F : an alphabet C: random code of rate r of F n Our Result lim Pr(δ H(C) > δ) = n 1, if r < 1 2 GV q(δ) ; 0, if r > 1 2 GV q(δ). Is δ H (C) > δ? YES 1 2 GV q(δ) r increasing r decreasing 1 2 GV q(δ) NO Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

84 Pictures for Phase Transitions: Linear case Linear case: random linear code L p 6 1 GV q(δ) - r Figure: p = Pr ( δ H (L) > δ ) ; 0 < δ < δ 0. Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

85 Pictures for Phase Transitions: Linear case Linear case: random linear code L r 6 1 GV-bound I: upper bound II: small large probability III: zero probability probability δ 0 - δ Figure: Three areas for the probability of the event δ H (L) > δ Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

86 Pictures for Phase Transitions: Linear case p 6 p = Pr(δ H (L) > δ) - r δ r = GV q (δ) is a phase transition curve Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

87 Pictures for Phase Transitions: Arbitrary case Arbitrary case: arbitrary random code C r 6 1 upper bound + GV-bound + I II II II III half GV-bound δ 0 - δ I: probability 1 II: probability 0 II : greedy algorithm works II : greedy algorithm does t work III: probability = 0 Figure: Three (four) areas for the probability of the event δ H (C) > δ Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36

88 Phase Transitions of Random Codes and GV-Bounds THANK YOU Y. Fan (CCNU) Phase Transitions of Random Codes and GV-Bounds Oct / 36