Block-transitive algebraic geometry codes attaining the Tsfasman-Vladut-Zink bound

Transcription

1 Block-transitive algebraic geometry codes attaining the Tsfasman-Vladut-Zink bound María Chara*, Ricardo Podestá**, Ricardo Toledano* * IMAL (CONICET) - Universidad Nacional del Litoral ** CIEM (CONICET) - Universidad Nacional de Córdoba CIMPA Research Skol Algebraic methods in Coding Theory July 2-15, 2017 / Ubatuba, São Paulo, Brazil.

2 Based on the joint work María Chara, Ricardo Podestá, Ricardo Toledano Block-transitive algebraic geometry codes attaining the Tsfasman-Vladut-Zink bound. Asymtotically good 4-quasi transitive algebraic geometry codes over rime fields, arxiv: v1 [math.nt]

3 Summary of the talk 1 1. Codes 2 2. Asymtotics 3 3. Good BTC from towers 4 4. GBTC from class field towers 5 5. GBTC over rime fields

4 Motivation Oen question Is the family of cyclic codes asymtotically good?

5 Block-transitive codes

6 Linear codes A linear code over F q of length n, dimension k and minimun distance d is F q -linear subsace C F n q with k = dim C d = min{d(c, c ) : c, c C, c c } where d is the Hamming distance in F n q. C is an [n, k, d]-code over F q.

7 Bounds Singleton bound Griesmer bound k + d n 1 k 1 d n q i i=0 Hamming and Gilbert bounds [ d 1 2 ] ( n ) i (q 1) i i=0 d 1 q n k ( n ) i (q 1) i i=0

8 Transitive and cyclic codes The ermutation grou S n acts naturally on F n q π (v 1,..., v n ) = (v π(1),..., v π(n) ) The ermutation grou of C is Aut(C) = {π S n : π(c) = C} S n C is transitive if Aut(C) acts transitively on C, i.e. if for any 1 i < j n there is some π Aut(C) s.t. π(i) = j. C is cyclic if σ = (12 n) Aut(C), i.e. c = (c 1,..., c n 1, c n ) C σ(c) = (c n, c 1,..., c n 1 ) C

9 Block-by-block actions If n = m 1 + m m r we can consider v F n q divided into r blocks of lengths m i v = (v 1,1,..., v 1,m1 ;... ; v r,1,..., v r,mr ) There is a block-by-block action of S m1 S mr on F n q, π v = (v 1,π1 (1),..., v 1,π1 (m);... ; v r,πr (1),..., v r,πr (m)) where π = (π 1,..., π r ) S m S m.

10 Block-transitive codes (BTC) Definition A code C of length n = m 1 + m m r is said to be block-transitive if for some r N there is a subgrou = {(π 1,..., π r )} < S m1 S mr acting transitively on the corresonding blocks in which the words of C are divided. If m 1 = m 2 = = m r = m, hence n = rm, we say that C is an r-block transitive code. If π 1 = = π r = π we have an r-quasi transitive code.

11 Algebraic geometric codes

12 AG-codes: definition We will use the language of algebraic function fields. Let F be an algebraic function field over F q. Let D = P P n and G be disjoints divisors of F, where P 1,..., P n are different rational laces. The Riemann-Roch sace associated to G L(G) = {x F : (x) G} {0} The AG-code defined by F, D and G is C(D, G) = {( x(p 1 ),..., x(p n ) ) : x L(G) } (F q ) n where x(p i ) stands for the residue class of x modulo P i in the residual field F Pi = O Pi /P i.

13 AG-codes: arameters C(D, G) is an [n, k, d]-code with d n deg G and k = dim L(G) dim L(D G). If deg G < n then, by Riemann-Roch, k = dim L(G) deg G + 1 g where g is the genus of F. If also 2g 2 < deg G then k = deg G + 1 g.

14 Geometric block-transitive codes Question How can one construct (geometric) block-transitive codes?

15 Asymtotically good codes

16 Asymtotically good codes The information rate and relative minimum distance of an [n, k, d]-code C are R = k n and δ = d n The goodness of C is usually measured according to how big is 0 < R + δ < 1 A sequence {C i } i=0 of [n i, k i, d i ]-codes over F q is called asymtotically good over F q if lim su i k i d i > 0 and lim su > 0 n i i n i where n i as i. Otherwise the sequence is said to be bad.

17 Asymtotically good families Definition: A family of codes is asymtotically good if there is a sequence in the family which is asymtotically good. is asymtotically bad if there is no asymtotically good sequence in the family. Examles: Self-dual codes are asymtotically good. Transitive codes are asymtotically good. Quasi-cyclic grous are asymtotically good. BCH codes are asymtotically bad. Cyclic codes? We don t know.

18 The Ihara function The Ihara s function is A(q) = lim su g N q (g) g where N q (g) is the maximum number of rational laces that a function field over F q of genus g can have. By the Serre and Drinfeld-Vladut bounds c log q A(q) q 1 for some c > 0. For F q 2 one has A(q 2 ) = q 1

19 Manin s function Consider the ma ψ : {linear codes over F q } [0, 1] [0, 1] C (δ C, R C ) For δ [0, 1], consider the accumulation oints of ψ(c) in the line x = δ. Define α q (δ) to be the greatest second coordinate of these oints. Theorem The Manin s function α q : [0, 1] [0, 1] is continuous with α q (0) = 1, α q is decreasing on [0, 1 1 A(q) ] and, α q = 0 in [1 1 A(q), 1].

20 Asymtotic bounds for α q (δ) Singleton bound α q (δ) 1 δ Griesmer bound α q (δ) 1 q q 1 δ Hamming and Gilbert-Varshamov bounds 1 H q ( δ 2 ) α q(δ) 1 H q (δ) where H q : [0, 1 1 q ] R is the q-ary entroy function H q (x) = x log q (q 1) x log q (x) (1 x) log q (1 x) and H q (0) = 0.

21 Tsfasman-Vladut-Zink bound Theorem (Tsfasman-Vladut-Zink bound) Let q be a rime ower. If A(q) > 1 then for δ [0, 1 1/A(q)]. α q (δ) 1 δ 1 A(q) The TVZ-bound imroves the GV-bound over F q 2, for q 2 49.

22 (l, δ)-bounds Definition Let 0 < δ < l < 1 A sequence {C i } i=0 of [n i, k i, d i ]-codes over F q is said to attain a (l, δ)-bound over F q if lim su i k i d i l δ and lim su δ. n i i n i Examle (Tsfasman-Vladut-Zink bound) A sequence {C i } i=0 of codes over F q with A(q) > 1 attains the TVZ-bound over F q if it attains a (l, δ)-bound with l = 1 1 A(q)

23 Asymtotically good towers

24 Towers of function fields A sequence F = {F i } i=0 of function fields over F q is called a tower if F i F i+1 for all i 0. F i+1 /F i is finite and searable of degree > 1 for all i 1. F q is algebraically closed in F i for all i 0. g(f i ) for i. A tower F is recursive if there exist a sequence {x i } i=0 of transcendental elements over F q and H(X, Y ) F q [X, Y ] such that F 0 = F q (x 0 ) and F i+1 = F i (x i+1 ), H(x i, x i+1 ) = 0, i 0.

25 Parameters of towers Let F = {F i } i=0 be a tower of function fields over F q. The genus of F over F 0 is defined as γ(f) := lim i g(f i ) [F i : F 0 ] The slitting rate of F over F 0 is defined as ν(f) := lim i N(F i ) [F i : F 0 ] where N(F i ) the number of rational laces of F i

26 Asymtotic behavior of towers the limit of the tower F is λ(f) := lim i N(F i ) g(f i ) = ν(f) γ(f) Note that 0 λ(f) A(q) <. A tower F is called asymtotically good over F q if ν(f) > 0 and γ(f) < Otherwise is called asymtotically bad. Equivalently, F is asymtotically good if and only if λ(f) := lim i N(F i ) g(f i ) > 0 and F is called otimal over F q if λ(f) = A(q).

27 Asymtotically good codes from towers

28 Asymtotically good AG-codes from towers Proosition Let F = {F i } i=0 be a tower such that for each i 1 there are n i rational laces P (i) 1,..., P(i) n i in F i satisfying (a) n i as i, (b) for λ (0, 1) there exists i 0 s.t. g(f i ) n i λ for all i i 0, and (c) for each i > 0 there exists a divisor G i of F i disjoint from such that D i := P (i) P n (i) i deg G i n i s(i) where s : N R with s(i) 0 as i.

29 Asymtotically good AG-codes from towers Proosition (continued) Then, there exists a sequence {r i } i=m N such that F induces a sequence G = {C i } i=m of asymtotically good AG-codes of the form C i = C L (D i, r i G i ) attaining a (l, δ)-bound with l = 1 λ and 0 < δ < l.

30 Conditions for asymtotically good BT codes

31 Ramification Let E/F be a function field extension of finite degree. Let Q and P be laces of E and F, with Q P. e(q P) and f (Q P) the ramification index and the inertia degree of Q P. P slits comletely in E if e(q P) = f (Q P) = 1 for any lace Q of E lying over P (hence there are [E : F ] laces in E above P). P ramifies in E if e(q P) > 1 for some lace Q of E above P P is totally ramified in E if there is only one lace Q of E lying over P and e(q P) = [E : F ] (hence f (Q P) = 1). E/F is called b-bounded if for any lace P of F and any lace Q of E lying over P we have e(q P) 1 d(q P) b(e(q P) 1)

32 Ramification Let F = {F i } i=0 be a tower of function fields over F q. The ramification locus R(F) of F is the set of laces P of F 0 such that P is ramified in F i for some i 1. The slitting locus S(F) of F is the set of rational laces P of F 0 such that P slits comletely in F i for all i 1. A lace P of F 0 is totally ramified in the tower if for each i 1 there is only one lace Q of F i lying over P and e(q P) = [F i : F 0 ]. Definition A lace P of F 0 is absolutely µ-ramified in F (µ > 1) if for each i 1 and any lace Q of F i lying over P we have that e(r Q) µ for any lace R of F i+1 lying over Q.

33 Ramification The tower F is tamely ramified if for any i 0, any lace P of F i and any lace Q of F i+1 lying over P, the ramification index e(q P) is not divisible by the characteristic of F q. Otherwise, F is called wildly ramified. F has Galois stes if each extension F i+1 /F i is Galois. F is a b-bounded tower if each extension F i+1 /F i is a b-bounded Galois -extension where = char(f q ). If each extension F i /F 0 is Galois, F is said to be a Galois tower over F q.

34 Theorem 1: general conditions for existence Theorem Let F = {F i } i=0 be either a tamely ramified tower with Galois stes or a 2-bounded tower over F q with S(F) and R(F). Suose there are finite sets Γ and Ω of rational laces of F 0 such that R(F) Γ and Ω S(F) with 0 < g ɛt < r where g 0 = g(f 0 ), t = Γ, r = Ω and ɛ = 1 2 if F is tamely ramified or ɛ = 1 otherwise. If a lace P 0 R(F) is absolutely µ-ramified in F for some µ > 1 then there exists a sequence C = {C i } i=0 of r-block transitive AG-codes over F q attaining a (l, δ)-bound with l = 1 g 0 1+ɛt r. In articular, the Manin s function satisfies α q (δ) l δ. Moreover, the sequence C is defined over the Galois closure E of F with limit λ(e) > 1.

35 Block-transitive codes attaining the TVZ-bound

36 From wild towers { q 2i 1 (res. q 2i 1 1/2 ) if 1 i 2, q is odd (res. even), m i = q 3i 3 (res. q 3i 3 1/2 ) if i 3, q is odd (res. even). Theorem Let q > 2 be a rime ower. Then, there exists a sequence C = {C i } i=1 of r-block transitive codes over F q 2, with r = q2 q, attaining the TVZ-bound. Each C i is an [n i = rm i, k i, d i ]-code. By fixing 0 < δ < 1 q 2, we also have that d i δn i and k i {(1 δ)r (q + q i )}m i for each i 1, where the second inequality is non trivial if δ satisfies 0 < δ < 1 1 r (q + q i ).

37 Sketch of roof Consider the wildly ramified tower F = {F i } i=0 over F q 2 recursively defined by the equation which is otimal ([GS 96]). y q + y = F is a 2-bounded tower over F q 2, x q x q the ole P of x 0 in F 0 = F q 2(x 0 ) is totally ramified in F, so that P is absolutely q-ramified in F, at least q 2 q rational laces of F q 2 slit comletely in F and the ramification locus R(F) has at most q + 1 elements, i.e. q 2 q S(F) and R(F) q + 1.

38 Sketch of roof We are in the conditions of Theorem 1 with µ = q, ɛ = 1, g 0 = 0, r = q 2 q and t = q + 1. Therefore, there exists a sequence C = {C i } i N of r-block transitive AG-codes over F q 2 attaining a (l, δ)-bound with l = 1 g t r = 1 q q 2 q = 1 1 q 1, Thus, F attains the TVZ-bound over F q 2.

39 Good block transitive from class field towers

40 GBTC from olynomials Given n, m Z we ut { 1 if n m, ε n (m) = 0 if n m. For h F q [t], we define S 2 q(h) = {β F q : h(β) is a non zero square in F q }, S 3 q(h) = {β F q : h(β) is a non zero cube in F q }.

41 GBTC from olynomials Theorem (case q odd) Let q be an odd rime ower and let h F q [t] be a monic and searable olynomial of degree m such that it slits comletely into linear factors over F q. Suose there is a set Σ o S 2 q(h) such that u = Σ o > 0 and 2 2u m (u ε 2 (m)) < 3u Then, there exists a tamely ramified Galois tower F over F q with 4u limit λ(f) m 2 ε 2 (m) > 1. In articular, there exists a sequence of asymtotically good 2u-block transitive codes over F q, constructed from F, attaining an (l, δ)-bound, with. l = 1 m 3 + ε 2(m) 4u

42 GBTC from olynomials Theorem (case q even) Let q = 2 2s and let h F q [t] be a monic and searable olynomial of degree m such that it slits comletely into linear factors over F q. Suose there is a set Σ e S 3 q(h) such that v = Σ e > 0 and 2 3v m (v ε 3 (m)) < 2v 1 2 Then, there exists a tamely ramified Galois tower F over F q with limit λ(f 6v ) 2(m ε 3 (m)) 3 > 1. In articular, there exists a sequence of asymtotically good 3v-block transitive codes over F q, constructed from F, attaining an (l, δ)-bound with. l = 1 2m 5 + 2ε 3(m) 6v

43 Sketch of roof Let K = F q (x) and F = F q (x, y) given by the equation y 2 = h(x) = (x a 1 ) (x a m ) F /K is cyclic Galois of degree 2. The rational laces P a1,..., P am of K are totally ramified in F /K and no other laces than P a1,..., P am and P ramify in F /K. Moreover, P is totally ramified if m is odd. There are m + 1 ε 2 (m) laces in K totally ramified in F. The genus g(f ) = 1 2 (m 1 ε 2(m)).

44 Sketch of roof By Kummer s theorem, the lace P β = P x β in K, β Σ o, slits comletely into 2 rational laces of F for each β Σ o. Let P 0 be some P ai and let Q 0 be the only lace of F lying above P 0. Let Q 1,..., Q 2u be the rational laces of F lying over the laces P β with β Σ o and ut Since T = {Q 0 } and S = {Q 1,..., Q 2u } #{P P(K) : P ramifies in F } = m + 1 ε 2 (m) thus, by hyothesis, #{P P(K) : P ramifies in F } 2 + Σ + 2 n Σ

45 Sketch of roof By a result of [AM], the T -tamely ramified and S-decomosed Hilbert tower HS T of F is infinite. This means that there is a sequence F = {F i } i=0 fields over F q such that F 0 = F, of function H T S = and for any i 1 each lace in S slits comletely in F i, the lace Q 0 is tamely and absolutely ramified in the tower, F i /F i 1 is an abelian extension, [F i : F ] as i and F i /F 0 is unramified outside T. i=0 F i

46 Sketch of roof Then, we are in the situation of Theorem 1 with F 0 = F, Γ = T, Ω = S Also, g(f ) = 1 2 {m 1 ε 2(m)} < 2u = S Thus, by Theorem 1, the result follows.

47 Good block transitive codes over rime fields

48 Exlicit olynomials Proosition Let q = r be an odd rime ower. Suose that: there are 4 distinct elements α 1, α 2, α 3, α 4 F q such that / {α 1, α 2, α 3, α 4 } for 1 i 4 and consider α 1 i h(t) = (t + 1) 4 (t α i )(t α 1 ) F q [t], i=1 there is α F q such that h(α) = γ 2 0, γ F q. Then there exists a sequence of 4-block transitive codes over F q attaining a ( 1 8, δ)-bound with 0 < δ < 1 8. Proof. Take m = 9 and u = 2 in the revious Theorem and note that h(0) = 1 is a nonzero square in F q. i

49 Asymtotically good 4-block transitive AG-codes over F 13 It is easy to see that there is no searable olynomial over F 11 of degree 9 satisfying the required conditions. Examle 2, 3, 4, 5 F 13 satisfy the conditions of the roosition. We have h(t) = (t+1)(t 2)(t 7)(t 3)(t 9)(t 4)(t 10)(t 5)(t 8) h(11) = 3 = 4 2 in F 13. Thus, there are asymtotically good sequences of 4-block transitive codes over F 13 attaining a ( 1 8, δ)-bound.

50 Infinitely many rimes Consider a rime 29. By Fermat s little theorem 5 h(t) = (t + 1) (t k)(t k 2 ) F [t] k=2 has 9 different linear factors. h(a) is a nonzero square in F for a F By multilicativity of the Legendre symbol ( ) h(a) = 1. ( ) h(t) ( ) 5 = t+1 k=2 ( t k ) ( ) t k 2

51 Infinitely many rimes For 2 j 1 5 we have h( j) = ( (j 1)) 5 ( )( (j + k) (j + k 2 ) ) 0 k=2 By modularity: ( ) h( j) = = = ( ) 1 j 5 k=2 ) 5 ( 1 j ( 1 j k=2 ) 5 k=2 ( j+k )( j+k 2 ( )( j+k k ) ) 2 ( ) j+k 2 ( )( )( ) j+k k kj+1

52 Infinitely many rimes For instance, for j = 2 we have Thus, ( ) h( 2) ( ) h( 2) = ( ) 5 = 1 k=2 ( )( k+2 k )( ) 2k+1 ( ) ( ( ) ( ) ( ) )( ( ) ( ) ( ) ) ( ( ) ( ) ( ) )( ( ) ( ) ( ) ) and hence ( ) h( 2) ( )( = 1 5 ) ( ) 11

53 Infinitely many rimes For 37 we can take the 2 j 7 and we have ( ) ( )( )( ) h( 2) = ) ( )( )( )( ) = ( h( 3) ( h( 4) ( h( 5) ( h( 6) ( h( 7) ) ( = 1 ) ( = 1 ) ( = 1 ) ( = 1 )( )( )( )( )( ) )( )( )( )( 2 )( ) 29 )( ) 17 )( 11 )( 13 ( )( 19 )( ) 31 This reduces the search of α F such that h(α) = γ 2 F ), to the comutation of Legendre symbols.

54 Infinitely many rimes Proosition There are asymtotically good 4-block transitive AG-codes over F for infinitely many rimes. For instance, this holds for rimes of the form = 220k + 1 or = 232k + 1, k N. Proof. As before, for 37, consider the olynomial h(t) = (t + 1) 5 (t k)(t k 2 ) F [t] k=2 It suffices ) to find infinitely many rimes, such that = 1, for a given j. ( h( j)

55 Infinitely many rimes Consider j = 2. We look for rime numbers such that ( ) ( ) ( ) ( ) h( 2) = = 1. ( ) ( ) Since 1 = 1 if 1 (4) and 5 = 1 if ±1 (5), it is ( ) ( ) clear that if = 20k + 1 then 1 5 = 1. ( ) In this way, if = (20 11)k + 1, k N, then h( 2) = 1 by quadratic recirocity. By Dirichlet s theorem on arithmetic rogressions, there are infinitely many rime numbers of the form = 220k + 1, k N (the first being = 661).

56 muito obrigado!