Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth s second tower

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1 Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth s second tower arxiv:5.0630v [cs.it] 9 Nov 05 Olav Geil, Stefano Martin, Umberto Martínez-Peñas, Diego Ruano Abstract Asymptotically good sequences of ramp secret sharing schemes were given in [4] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW. Introduction Relative generalized Hamming weights (RGHWs) of two linear codes are fundamental for evaluating the security of ramp secret sharing schemes and wire-tap channels of type II [5, 6, 8, ]. Until few years ago only the RGHWs of MDS codes and a few other examples of codes were known [7], but recently new results were discovered for one-point algebraic geometric codes [5], q-ary Reed-Muller codes [9] and cyclic codes [3]. In [4] it was discussed how to obtain asymptotically good sequences of ramp secret sharing schemes by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. The tools used in [4] were the Goppa bound and the Supported by the Danish Council for Independent Research, grant DFF and the Spanish Ministry of Economy, grant MTM C Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 90-Aalborg Øst, Denmark. {olav,stefano,umberto,diego}@math.aau.dk

2 Feng-Rao bounds. In the present paper we focus on secret sharing schemes coming from the Garcia-Stichtenoth s second tower [3]. We give a method for obtaining new information on the RGHWs when the used codes have small codimension. For general codimension we give an improved estimate on the highest RGHW. The new results are obtained by studying in detail the sequence of Weierstrass semigroups related to the sequence of rational places [0]. We recall the definition of RGHWs and briefly mention their use in connection with secret sharing schemes. Definition. Let C C F n q be two linear codes. For m =,..., l = dim C dim C the m-th relative generalized Hamming weight of C with respect to C is M m (C, C ) = min{#suppd D C is a linear space, dim D = m, D C = { 0}}. Here SuppD = #{i N exists (c,..., c n ) D with c i 0}. Note that for m =,..., dim C, the m-th generalized Hamming weight (GHW) of C d m (C ) is M m (C, { 0}). Given C C linear codes, by definition, we have that the m-th generalized Hamming weight is a lower bound for the m-th relative generalized Hamming weight of C with respect to C, i.e. M m (C, C ) d m (C ). In [], a general construction of a linear secret sharing scheme with n participants is defined from two linear codes C C of length n. It was proved in [5, 6] that it has r m = n M l m+ (C, C )+ reconstruction and t m = M m (C, C ) privacy for m =,..., l. Here, r m and t m are the unique numbers such that the following holds: It is not possible to recover m q-bits of information about the secret with only t m shares, but it is possible with some t m + shares. With any r m shares it is possible to recover m q-bits of information about the secret, but it is not possible to recover m q-bits of information with some r m shares. We shall focus on one-point algebraic geometric codes C L (D, G) where D = P P n, G = µq, and P,..., P n, Q are pairwise different rational places over a function field. By writing ν Q for the valuation at Q, the Weiestrass semigroup corresponding to Q is ( ) H(Q) = ν Q L(µQ) = {µ N 0 L(µQ) L((µ )Q)}. µ=0 We denote by g the genus of the function field and by c the conductor of the Weierstrass semigroup. We consider C = C L (D, µ Q) and C =

3 C L (D, µ Q), with µ < µ. Observe that for l = dim(c ) dim(c ) and µ = µ µ we have that l µ, with equality if g µ < µ n holds. From [5, Theorem 9] we have the following bound: Theorem. For m =,..., l we have that: where M m (C, C ) n µ + Z(H(Q), µ, m), Z(H(Q), µ, m) = min{#{α m s= (i s + H(Q)) α / H(Q)} (µ ) i < i <... < i m }. For m > g, one has that d m (C) = n k + m, that is the Singleton bound is reached [, Corollary 4.]. For other values of m, using Theorem, the following result was found [4, Proposition 4]. Proposition 3. Let C L (D, µq) be a one-point algebraic geometric code of length n and dimension k. If µ < n and m min{k, g}, then: d m (C L (D, µq)) n k + m c + h c m n k + m c where h c m = #(H(Q) (0, c m]). Moreover, in the proof of [4, Proposition 4], one has that d m (C L (D, µq)) n µ + g + m c + h c m, which may allow us to improve the bound in Proposition 3 for µ g, since in this case k µ + g. Furthermore, we can apply it to bound the RGHWs of a pair of codes. Proposition 4. Let C L (D, µ Q) C L (D, µ Q) be two one-point algebraic geometric codes of length n and dimension k and k, respectively. If µ < µ < n and m min{k, g} then d m (C L (D, µ Q)) n µ + g + m c + h c m where h c m = #(H(Q) (0, c m]). Moreover, if m min{k k, g} then M m (C L (D, µ Q), C L (D, µ Q)) n µ + g + m c + h c m 3

4 From Garcia-Stichtenoth s second tower [3] one obtains codes over any field F q where q is an even power of a prime. Garcia and Stichtenoth analyzed the asymptotic behaviour of the number of rational places and the genus, from which one has that the codes beat the Gilbert-Varshamov bound for q 49. This allows us to create sequences of asymptotically good codes. The Garcia-Stichtenoth s tower (F, F, F 3,...) in [3] over F q, for q an even power of a prime, is given by: F = F q (x ) for ν >, F ν = F ν (x ν ) with x ν satisfying x q ν + x ν = x q ν q x +. ν The number of rational points of F ν is N q (F ν ) q ν (q q) and its genus is g ν = g(f ν ) = (q ν+ )(q ν+ ). For every function field F ν the following complete description of the Weierstrass semigroups corresponding to a sequence of rational places was given in [0]. Let Q ν be the rational point that is the unique pole in x. The Weierstrass semigroups H(Q ν ) at Q ν in F ν are given recursively by: H(Q ) = N 0 H(Q ν ) = q H(Q ν ) {i N 0 : i c ν }, where c ν = q ν q ν+ is the conductor of H(Qν ). An alternative way to obtain these Weiestrass semigroups was described in []. Definition 5. First we define H(Q ) = N 0. j = ν, we define: For ν positive integer and c ν = q ν q (ν j ), H(Q ν ) = S 0 ν S ν S ν... S j ν S ν, where: S 0 ν = {x ν, } = {0} For i j, Si ν = {x i, x ν,q + ν,q i, x + ν,q i,..., x +3 ν,q i } where for k q i q i we have that x i = q j q ν i+ + kq ν i+ ν,q +k S ν = [c ν +, ) Using the previous description of the Weierstrass semigroup H(Q ν ), we can see that it has the following properties: 4

5 Lemma 6. With the same notation as before, one has that:. For any i, i {0,,,..., j, j, }, i i, we have that S i ν S i ν =.. For i {,..., j } we have that #Si ν = q i q i and # ( i r=0s r ν) = q i. 3. For i {,..., j } and for any two consecutive elements x, y Si ν, with x > y, we have that x y = q ν i+. 4. For i {,..., j }. Let x be the first element of Si ν and y the last element of Sν i, we have that x y = q ν i+. 5. For i {,..., j }, and for any x, y i r=0s r ν, x > y we have that x y q ν i+. Proof.. By Theorem in [].. Let i {,..., j }, the cardinality of Si ν follows by its definition. For the second part, by (), we have that: # ( i r=0s r ν) = #S 0 ν + i #Sν r = + r= i r= (q r q r ) = + q i = q i 3. Consider two consecutive elements x, y Sν, i x > y. There exists a k {,..., q i q i } such that x = q j q ν i+ + kq ν i+ and y = q j q ν i+ + (k + )q ν i+. It follows that x y = q ν i+. 4. Let y be the last element of Sν i, i.e. y = q j q ν i+, and x be the first element of Sν, i i.e. x = q j q ν i+ + q ν i+. We have that x y = q ν i+. 5. For i {,..., j }, consider x, y i r=0sν r with x > y. This mean that there exists i, i {,..., i}, i i such that x S i ν, y S i ν. Let x be the element that precedes x in H(Q ν ), then we have that x y = (x x ) + (x y) = q ν i + + (x y) q ν i + q ν i+. The second inequality follows by (3) and (4), the third one since x y and the last one since i i.. Applying Proposition 3 to code pairs coming from Garcia-Stichtenoth s second tower [3], an asymptotic result was given in [4, Theorem 3], which combined with Proposition 4 allows us to obtain the following result. 5

6 Corollary 7. Let (F i ) i= be Garcia-Stichtenoth s second tower of function fields over F q, where q is an even power of a prime. Let (C i ) i= be a sequence of one-point algebraic geometric codes constructed from (F i ) i=. Consider R, R, ρ with 0 R q, 0 R < and 0 ρ min{r, q }, and assume that dim(c i )/n i R and µ i /n i R. For all sequences of positive integers (m i ) i= with m i /n i ρ, it holds that δ = lim inf i d m (C i )/n i satisfies δ R + ρ. q () δ R + ρ. () Note that the bound () is sharper than () for q R q. Small codimension In this section we give a refined bound on the RGHWs of two nested one-point algebraic geometric codes coming from Garcia-Stichtenoth s towers when the codimension is small. Before giving such bound, we illustrate the main idea with an example. Example. Consider q = 9 and let F 6 be the 6-th function field defined by the Garcia-Stichtenoth s tower over F q. The Weierstrass semigroup H(Q 6 ) at Q 6 in F 6 is H(Q 6 ) = {0, 43, 486, 53, 540, 567, 594, 6, 648} {3n n N and 3n [654, 70]} {n N n > 70}. We denote these three sets as A 0, B 0 and C 0 respectively. For computing Z(H(Q 6 ), µ, m) one should find i,..., i m such that (µ ) i < i < < i m and minimize #{α m s= (i s + H(Q 6 )) α / H(Q 6 )}. In this example we fix i = 0, thus: i + H(Q 6 ) = { 0, 3, 466, 493, 50, 547, 574, 60, 68} {3n 0 n N and 3n [654, 70]} {n N n > 68} = = (i + A 0 )f (i + B 0 ) (i + C 0 ). Note that i +A 0 and H(Q 6 ) are disjoint since i = 0 < 7 and x y 7 for any x, y A 0 x y. For the same reason for any 0 < i < < i m, we have that i m +A 0, i m +A 0,..., i +A 0, i +A 0 and H(Q 6 ) 6

7 are disjoint. It follows that m v= (i v + A 0 ) {α m v= (i v + H(Q 6 )) α / H(Q 6 )} and # m v= (i v +A 0 ) = m v= #(i v +A 0 ) = (m )#A 0 = 9(m ). The same argument does not hold for i + B 0 (or i + C 0 ) because there exists x, y B 0 (or C 0 ) such that x y = 3 and i > 3 thus it is possible that (i v + B 0 ) B 0 (or (i v + C 0 ) C 0 ) for some v =,..., m. Note that #((i +C 0 )\C 0 ) = i = 0, but (i +C 0 )\C 0 may intersect A 0 B 0. Therefore we consider (i + C 0 )\ qn, in this way (i + C 0 )\(C 0 qn) and H(Q 6 ) are disjoint. It follows that if i = 0, then Z(H(Q 6 ), µ, m) # m v= (i v + A 0 ) + #((i + C 0 )\(C 0 qn)) = (m ) As we can see from previous example, we do not consider the sets B 0, i +B 0,..., i m +B 0 because of their intersections with other sets. In general, we will also consider all the possible values i in the range [m, µ ] to obtain the following bound. Theorem 8. Let ν be an even positive integer and q an even power of a prime. Consider two one-point algebraic geometric codes C = C L (D, µ Q) C = C L (D, µ Q) of length n built on the ν-th Garcia-Stichtenoth s function field over F q and µ = µ µ. For µ < q ν+ ( ( )), m =,..., µ, consider u = + ν + log m 3 4 q ( and β = min{q ν+ 4 ( q )(µ ) 3 + q ), q ν 5 ( q ) + }, we have that: 4 M m (C, C ) n µ + g(m) where min { (m )q ν 4 u + q u ( q ) : if m > β u {log g(m) = q (m ), log q(µ ) + 3}} (m )q ν 4 u + q u ( q ) if m β, m 0 if m = Proof. By Theorem we have that M m (C, C ) n µ + Z(H(Q ν ), µ, m) thus we will estimate Z(H(Q ν ), µ, m). If m =, Z(H(Q ν ), µ, ) = 0, otherwise we denote the conductor of H(Q ν ) by c. Set (µ ) i < < i m, we define u(i ) = log q ( i ) +, then q u(i ) i < q u(i )+. For the sake of simplicity we write u instead of u(i ). To estimate Z(H(Q ν ), µ, m) we consider the following two sets: A(i, i,..., i v ) = {α m v= (i v + A 0 (u)) α / H(Q ν )} where A 0 (u) = ν u i=0 Si ν and C(i ) = (i + C 0 )\H(Q ν ) where C 0 = {α N α > c}. Again, to simplify the notation we write A = A(i, i,..., i v ), A 0 = A 0 (u) and C = C(i ). By 7

8 construction A C {α m s= (i s + H(Q ν )) α / H(Q ν )} and A C =. Thus we have that Z(H(Q ν ), µ, m) #A + #C. We start by computing the cardinality of A. By definition of A 0 for any x, y A 0, x y there exist i x, i y {0,..., ν u} such that x Six ν and y Sν iy. We can assume without loss of generality that i x i j and x > y, then we obtain by (6) in Lemma 6 that x y q ν ix+. Since µ < q ν+, then i x ν ν ix+ u 0 and x y q q ν ( ν u)+ = q u+. Thus for x, y A 0, x > y, we have that x y q u+. Since i < q u+, it follows that (j + A 0 ) (j + A 0 ) = for any j, j [i, 0]. Therefore we have that #A = # m v= (i v +A 0 ) = (m )#A 0. By () in Lemma 6, we have #A 0 = #( ν u i=0 Si ν) = q ν 4 u. Thus #A = (m )q ν 4 u. Furthermore, #C = #((i +C 0 )\H(Q ν )) = #([c+i, c)\h(q ν )) #([c+ i, c)\ qn) = i ( q ) where the inequality follows since H(Q ν ) [0, c) qn. Hence, M m (C, C ) n µ + Z(H(Q ν ), µ, m) n µ + min (#A + #C) i { (µ ),..., (m )} n µ + min { ) (m )q ν 4 u + i ( q i { (µ ),..., (m )} }. One could try to minimize the previous expression bounding u by log q ( i )+. However, the obtained bound is too loose. Hence, we consider the minimum among all possible values of u instead of i : ( ) M m (C, C ) n µ + min {(m )q ν4 u + q u q { u log q (m ) +, log q (m) +,..., log q (µ ) + } } { ( ) n µ + min (m )q ν 4 u + q u q { u log q (m ) +, log q (m) +,..., log q (µ ) + } } { ( ) n µ + min (m )q ν 4 u + q u q log q (m ) u log q(µ ) + }, 8

9 where the second to last inequality is obtained since i q u ( ). We define f(u) = (m )q ν 4 u + q u q. In this way our bound becomes: { M m (C, C ) n µ +min f(u) log q (m ) u log q(µ ) + }. By looking( the derivative( of f(u), )) one can see that f(u) only has a minimum at u = + ν + log m 3 4 q (. However, it does not always hold that q ) log q (m ) / u log q (µ ) + /. This happens when either u < log q (m ) or u > log q (µ ) +. The first case is equivalent to m > q ν 5 ( q ) +, the second one to m > q ν+ 4 ( q )(µ ) +. 4 Thus if m > β = min{q ν+ 4 ( q )(µ ) 3 +, q ν 5 ( q ) + }, 4 then the minimum is reached in log q (m ) or log q(µ ) +. The previous result has an asymptotic implication as well. Corollary 9. Let q be an even power of a prime, 0 R R <, and R = R R < q. There exists a sequence of pairs of one-point AG codes C,i = C L (D i, µ,i Q) C,i = C L (D i, µ,i Q), such that: n i = n(c,i ) = n(c,i ), µ j,i /n i R j when i, for j =,. For a given ρ let m i be such that m i /n i ρ when i and let M = lim inf M mi (C,i, C,i )/n i. It holds that: M R + g(ρ), where min w {ρ, R} {ρ(w(q q)) + w(q } q) if ρ > β, q ( ) g(ρ) = ρ 3 ( + ρ ) 3 q q if ρ β, ρ 0, 0 if ρ = 0, and β = min { 4 q 5 ( q ) 3, q 4 (R } q R) 3. Proof. Consider the Garcia-Stichtenoth s tower (F, F,...) over F q described at the end of section, and 0 µ,i < µ,i n i with µ j,i /n i R j for j =,. Now consider C j,i = C L (D i, µ j,i Q) for j =,, where D i is a divisor of degree n i and with n i distinct places not containing Q i, which is the unique pole of x F i. By taking the limit of the bound obtained in Theorem 8, the corollary holds. 9

10 Note that if we assume that C,i is the zero code for all i, then lim inf M mi (C,i, { 0}) is the asymptotic value of the m i -th general Hamming weight of C i,. For R < 4(q, the bound in Corollary 9 is sharper than the one obtained in q) [4, Theorem 3]. In the following graph we compare the bound from Corollary 7 (the dashed curve) with the bound from Corollary 9 (the solid curve). The first axis represents ρ = lim m i /n i, and the second axis represents δ = lim inf M mi (C,i, { 0}). 3 The highest RGHW As we illustrated at the beginning of section, for any n M l (C, C ) + obtained shares an eavesdropper may recover at least one q-bit of the secret. In this section, for g µ < µ n, we obtain a refined bound for the highest RGHW of two one-point algebraic geometric codes obtained from Garcia-Stichtenoth s towers, i.e. M l (C, C ). Proposition 0. Let ν be an even positive integer and g µ < µ n. Consider two one-point algebraic geometric codes C = C L (D, µ Q) C = C L (D, µ Q) built on the ν-th Garcia-Stichtenoth s tower. We have that µ = µ µ = dim(c ) dim(c ) = l and M µ (C, C ) n dim C M µ (C, C ) n dim C if µ q ν, ( ν+ q ν log q (µ) (q i q i )+ i= 0

11 +(q ν+ ν+ log q (µ) µ)q ν+ log q (µ) ) if µ < q ν. Proof. Since g µ < µ n, then µ = µ µ = l [4, Lemma ]. By Theorem, we have that M µ (C, C ) n µ + Z(H(Q), µ, µ), where Z(H(Q), µ, µ) = #{α µ v=( v + H(Q)) α / H(Q)}. For any x, y H(Q), we have that x y q ν, thus if µ q ν then ( µ v=0 v+h(q)) = N 0 {,..., (µ )}. It follows that Z(H(Q), µ, µ) = (N 0 \H(Q ν )) {,..., (µ )}) = g + µ. Thus M µ (C, C ) n µ + g + µ. Moreover since µ g, then µ g + = dim C and the first part of the proposition holds. For l q ν we claim that: Z(H(Q), µ, µ) = µ + g (q ν u (µ) i= (q i + q i ) + u (µ)q u (µ) ), where u (µ) = ν+ log q (µ) ν u (µ)+ and u (µ) = q µ. This means that µ = q ν+ u(µ) u (µ). We prove it by decreasing induction on µ, for q ν µ. For the basis step we have µ = q ν, thus u (µ) = and u (µ) = 0. According to our claim, Z(H(Q), µ, µ) is equal to µ + g, which has been already proven in the first part of this proposition. For the inductive step, we now assume that our claim is true for Z(H(Q), µ, µ) and we want to prove it for Z(H(Q), µ, µ ). We note that: Z(H(Q), µ, µ) = #{α µ v=( v + H(Q)) α / H(Q)} + #{α (µ ) + H(Q)) α / µ v=0( v + H(Q))} = Z(H(Q), µ, µ ) + #T (µ), where T (µ) = {α (µ ) + H(Q) α / µ v=0( v + H(Q))}. Thus Z(H(Q), µ, µ ) = Z(H(Q), µ, µ) #T (µ). We consider two cases: µ such that q ν u (µ ) < µ < q ν u (µ )+ and µ = q ν u (µ )+ Let us consider the first case, µ such that q ν u (µ ) < µ < q ν u (µ )+ then u (µ ) = u (µ) and u (µ ) = u (µ) +. By induction we have that Z(H(Q), µ, µ) = µ+g (q ν u (µ), i= (q i q i ) + u (µ)q u (µ) ). We claim that #T (µ) = q u (µ). By (5) in Lemma 6, for any x, y u (µ ) i=0 Sν, i x > y we have that x y q ν u (µ )+, moreover µ < q ν u (µ )+. Therefore, one has that ( (µ ) + u (µ ) i=0 Sν) i

12 ( µ v=0 v + H(Q)) =. Then (µ ) + u (µ ) i=0 Sν i T (µ). Actually, the previous inclusion is an equality. We shall prove it by contradiction: we assume that there exists an element x T (µ) but not in (µ ) + u (µ ) i=0 Sν. i By definition of T (µ), we have that x (µ ) + (Sν ( j/ i=u (µ )+ Si ν)) where j = ν/. Consider y < x to be the previous element of x in (µ ) + H(Q). By (3) and (4) in Lemma 6, we have that x y q ν u (µ ) < µ. Thus, (µ ) (µ ) + (x y) 0 and x (µ ) + (x y) + H(Q) µ v=0( v + H(Q)). This means x / T (µ), which is a contradiction. It follows that #T (µ) = # ( (µ ) + u ) ( (µ ) i=0 Sν i u = # (µ ) ) i=0 Sν i u (µ ) = q = q u (µ). We consider now the second case, µ = q ν u (µ )+, then u (µ ) = u (µ)+ and u (µ ) = 0. By using the same argument as in the first case, one may also prove that #T (µ) = q u (µ). Corollary. By using the same notation of the previous proposition, for g µ < µ < n and l q ν we have that M l (C, C ) = n dim C. Proof. By 0, M l (C, C ) n dim C. And M l (C, C ) n dim C, by the Singleton bound for one-point algebraic geometric codes and the result holds. This means that for l q ν the Singleton bound is reached. Note that for l < q ν, the bound in Proposition 0 allows us to get a refined bound since we could consider h c m. As before, this result has an asymptotically implication: Corollary. Let q be an even power of a prime, q R R <, and R = R R. There exists a sequence of one-point algebraic geometric codes C,i = C L (D i, µ,i Q) C,i = C L (D i, µ,i Q), µ i = µ,i µ,i, such that: n i = n(c,i ) = n(c,i ), µ j,i /n i R j when i, for j =,. Let l i = dim C,i dim C,i, M = lim inf M li (C,i, C,i )/n i, R j = lim dim C i,j n i for j =,, and R = R R, we have that: M = R if R q q

13 and ( M R q q ) +q + log q (R( q )) ( logq (R( q )) i= + Rq log q (R( q )) ) (q i q i )+ if R < q q. Proof. Let (F, F,...) be the tower of function fields defined in section, and 0 µ,i < µ,i n i with µ j,i /n i R j for j =,, where n i is the length of rational places of F i. Now consider C j,i = C L (D i, µ j,i Q) for j =,, where D i is a divisor of degree n i, with n i distinct places not containing Q, which is the unique pole of x F i. Since we assume that q R R < then R j = R j q, for j =, and R = R. By taking the limit of the result obtained in Proposition 0 and Corollary the result holds. Note that if log q (R( q )) = log q (R( q )) then the formulas in Corollary become: and M R q q M = R if R log q (R( q )) i= q q ( ) q i q i if R < q q. Corollary 7 can be used for ρ min{r, q }. If C,i = {0} for all i, then the value M of Corollary represents the asymptotic value of the highest GHW of C i,. Note that Corollary can be used for any value of R, but 7 cannot. References [] Maria Bras-Amorós and Michael E O Sullivan. The order bound on the minimum distance of the one-point codes associated to the Garcia Stichtenoth tower. Information Theory, IEEE Transactions on, 53():44 445,

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