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

Asymptotically good sequences of ramp secret sharing schemes were given in [Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] 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 [5,6,8,12]. 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 [13]. 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 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 [10].
We recall the definition of RGHWs and briefly mention their use in connection with secret sharing schemes. Definition 1. Let C 2 C 1 ⊆ F n q be two linear codes. For m = 1, . . . , = dim C 1 − dim C 2 the m-th relative generalized Hamming weight of C 1 with respect to C 2 is Here SuppD = #{i ∈ N | exists (c 1 , . . . , c n ) ∈ D with c i = 0}. Note that for m = 1, . . . , dim C 1 , the m-th generalized Hamming weight (GHW) of C 1 Given C 2 C 1 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 1 with respect to C 2 , i.e. M m (C 1 , C 2 ) ≥ d m (C 1 ). In [2], a general construction of a linear secret sharing scheme with n participants is defined from two linear codes C 2 C 1 of length n. It was proved in [5,6] that it has r m = n−M −m+1 (C 1 , C 2 )+1 reconstruction and t m = M m (C ⊥ 2 , C ⊥ 1 )−1 privacy for m = 1, . . . , . 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 + 1 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 − 1 shares.
We shall focus on one-point algebraic geometric codes C L (D, G) where D = P 1 + . . . + P n , G = µQ, and P 1 , . . . , 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 We denote by g the genus of the function field and by c the conductor of the Weierstrass semigroup. We consider C 1 = C L (D, µ 1 Q) and C 2 = From [5,Theorem 19] we have the following bound: Theorem 2. For m = 1, . . . , we have that: For m > g, one has that d m (C) = n − k + m, that is the Singleton bound is reached [11,Corollary 4.2]. For other values of m, using Theorem 2, the following result was found [4,Proposition 14].
Proposition 3. Let C L (D, µQ) be a one-point algebraic geometric code of length n and dimension k. If −1 ≤ µ < n and 1 ≤ m ≤ min{k, g}, then: Moreover, in the proof of [4,Proposition 14], one has that which may allow us to improve the bound in Proposition 3 for µ ≤ 2g − 2, since in this case k ≥ µ + 1 − g. Furthermore, we can apply it to bound the RGHWs of a pair of codes.
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 1 , F 2 , F 3 , . . .) in [3] over F q , for q an even power of a prime, is given by: .

The number of rational points of
. For every function field F ν the following complete description of the Weierstrass semigroups corresponding to a sequence of rational places was given in [10]. Let Q ν be the rational point that is the unique pole in x 1 . The Weierstrass semigroups H(Q ν ) at Q ν in F ν are given recursively by: is the conductor of H(Q ν ). An alternative way to obtain these Weiestrass semigroups was described in [1].
Definition 5. First we define H(Q 1 ) = N 0 . For ν positive integer and j = 2 ν 2 , we define: where: Using the previous description of the Weierstrass semigroup H(Q ν ), we can see that it has the following properties: Lemma 6. With the same notation as before, one has that: 3. For i ∈ {1, . . . , j 2 } and for any two consecutive elements x, y ∈ S i ν , with 2. Let i ∈ {1, . . . , j 2 }, the cardinality of S i ν follows by its definition. For the second part, by (1), we have that: 3. Consider two consecutive elements x, y ∈ S i ν , x > y. There exists a k ∈ {1, . . . , q , and x be the first element of . The second inequality follows by (3) and (4), the third one since x 2 ≥ y and the last one since i 1 ≤ i.
Applying Proposition 3 to code pairs coming from Garcia-Stichtenoth's second tower [3], an asymptotic result was given in [4,Theorem 23], which combined with Proposition 4 allows us to obtain the following result.
Note that the bound (2) is sharper than (1)

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. 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 1 , . . . , i m−1 such that −(µ− 1) ≤ i 1 < i 2 < · · · < i m−1 ≤ −1 and minimize #{α ∈ ∪ m−1 s=1 (i s + H(Q 6 )) | α / ∈ H(Q 6 )}. In this example we fix i 1 = −20, thus: Note that i 1 +A 0 and H(Q 6 ) are disjoint since −i 1 = 20 < 27 and |x−y| ≥ 27 for any x, y ∈ A 0 x = y. For the same reason for any −20 < i 2 < · · · < i m−1 ≤ −1, we have that i m−1 + A 0 , i m−2 + A 0 , . . . , i 2 + A 0 , i 1 + A 0 and H(Q 6 ) . 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 1 > 3 thus it is possible As we can see from previous example, we do not consider the sets B 0 , i 1 +B 0 , . . ., i m−1 +B 0 because of their intersections with other sets. In general, we will also consider all the possible values −i 1 in the range [m − 1, µ − 1] 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 2 = C L (D, µ 2 Q) C 1 = C L (D, µ 1 Q) of length n built on the ν-th Garcia-Stichtenoth's function field over F q and µ = µ 1 − µ 2 . For µ < q ν+1 2 , m = 1, . . . , µ, consider u * = 2 For the sake of simplicity we write u instead of u(i 1 ).
To estimate Z(H(Q ν ), µ, m) we consider the following two sets: Again, to simplify the notation we write A = A(i 1 , i 2 , . . . , i v ), A 0 = A 0 (u) and C = C(i 1 ). By 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 2 − u} such that x ∈ S ix ν and y ∈ S iy ν . We can assume without loss of generality that i x ≥ i j and x > y, then we obtain by (6) One could try to minimize the previous expression bounding u by log q (−i 1 )+ 1 2 . However, the obtained bound is too loose. Hence, we consider the minimum among all possible values of u instead of i 1 : where the second to last inequality is obtained since −i 1 ≥ q u− 1 2 . We define In this way our bound becomes: By looking the derivative of f (u), one can see that f (u) only has a minimum at u * = 2 . However, it does not always hold that log q (m − 1) − 1/2 ≤ u * ≤ log q (µ − 1) + 1/2. This happens when either u * < log q (m − 1) − 1 2 or u * > log q (µ − 1) + 1 2 . The first case is equivalent to then the minimum is reached in log q (m − 1) − 1 2 or log q (µ − 1) + 1 2 . The previous result has an asymptotic implication as well.
Corollary 9. Let q be an even power of a prime, 0 ≤R 2 ≤R 1 < 1, and There exists a sequence of pairs of one-point AG codes For a given ρ let m i be such that m i /n i → ρ when i → ∞ and let M = lim inf M m i (C 1,i , C 2,i )/n i . It holds that: Proof. Consider the Garcia-Stichtenoth's tower (F 1 , F 2 , . . .) over F q described at the end of section 1, and 0 ≤ µ 2,i < µ 1,i ≤ n i − 1 with µ j,i /n i →R j for j = 1, 2. Now consider C j,i = C L (D i , µ j,i Q) for j = 1, 2, where D i is a divisor of degree n i − 1 and with n i − 1 distinct places not containing Q i , which is the unique pole of x 1 ∈ F i . By taking the limit of the bound obtained in Theorem 8, the corollary holds.
Note that if we assume that C 2,i is the zero code for all i, then lim inf M m i (C 1,i , { 0}) is the asymptotic value of the m i -th general Hamming weight of C i,1 . For R < 1 4(q− √ q) , the bound in Corollary 9 is sharper than the one obtained in [4,Theorem 23].
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 m i (C 1,i , { 0}).

The highest RGHW
As we illustrated at the beginning of section 1, for any n − M (C 1 , C 2 ) + 1 obtained shares an eavesdropper may recover at least one q-bit of the secret. In this section, for 2g − 1 ≤ µ 2 < µ 1 ≤ n − 1, 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 (C 1 , C 2 ).
By (3) and (4) in Lemma 6, we have that Proof. Let (F 1 , F 2 , . . .) be the tower of function fields defined in section 1, and 0 ≤ µ 2,i < µ 1,i ≤ n i − 1 with µ j,i /n i →R j for j = 1, 2, 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 = 1, 2, where D i is a divisor of degree n i −1, with n i −1 distinct places not containing Q, which is the unique pole of x 1 ∈ F i . Since we assume that 2 √ q−1 ≤R 2 ≤R 1 < 1 then R j =R j − 1 √ q−1 , for j = 1, 2 and R =R. By taking the limit of the result obtained in Proposition 10 and Corollary 11 the result holds.
Note that if log q (R(1 − 1 √ q )) = log q (R(1 − 1 √ q )) then the formulas in Corollary 12 become: Corollary 7 can be used for ρ ≤ min{R, 1 √ q−1 }. If C 2,i = {0} for all i, then the value M of Corollary 12 represents the asymptotic value of the highest GHW of C i,1 . Note that Corollary 12 can be used for any value of R, but 7 cannot.