Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric codes [4]-[8], [10]. In those works the focus is on full privacy and full reconstruction. In this paper we analyze additional parameters describing the asymptotic behavior of partial information leakage and possibly also partial reconstruction giving a more complete picture of the access structure for sequences of linear ramp secret sharing schemes. Our study involves a detailed treatment of the (relative) generalized Hamming weights of the considered codes.

It is a well-known fact that the BMS algorithm with majority voting can decode up to half the Feng-Rao designed distance dFR. Since dFR is not smaller than the Goppa designed distance dG, that algorithm can correct up to errors. On the other hand, it has been considered to be evident that the original BMS algorithm (without voting) can correct up to errors similarly to the basic algorithm by Skorobogatov-Vladut. But, is it true? In this short paper, we show that it is true, although we need a few remarks and some additional procedures for determining the Groebner basis of the error locator ideal exactly. In fact, as the basic algorithm gives a set of polynomials whose zero set contains the error locators as a subset, it cannot always give the exact error locators, unless the syndrome equation is solved to find the error values in addition.

List decoding is a process by which a list of decoded words is output instead of one. This works for a larger noise threshold than the traditional algorithms. Under some circumstances it becomes useful to be able to find out the actual message from the list. List decoding is assumed to be successful, meaning, the sent message features in the decoded list. This problem has been considered by Guruswami. In Guruswami's work, this disambiguation is done by sending supplementary information through a costly, error-free channel. The model is meaningful only if the number of bits of side information required is much less than the message size. But using deterministic schemes one has to essentially send the entire message through the error free channel. Randomized strategies for both sender and receiver reduces the required number of bits of side information drastically. In Guruswami's work, a Reed-Solomon code based hash family is used to construct such randomized schemes. The scheme with probability utmost ε reports failure and returns the whole list. The scheme doesn't output a wrong message. Also, in Guruswami's work some theoretical bounds have been proved which lower bound the bits of side information required. Here we examine whether the gap between the theoretical bounds and existing schemes may be narrowed. Particularly, we use the same scheme as in Guruswami's work, but use hash families based on Hermitian curve and function fields of Garcia-Stichtenoth tower and analyze the number of bits of side information required for the scheme.

In this paper, we present a lower bound for the dimension of subfield subcodes of residue Goppa codes on the curve Cab, which exceeds the lower bound given by Stichtenoth when the number of check symbols is not small. We also give an illustrative example which shows that the proposed bound for the dimension of certain residue Goppa code exceeds the true dimension of a BCH code with the same code length and designed distance.

In this paper, we give a new lower bound for the dimension of subfield subcodes. This bound improves the lower bound given by Stichtenoth. A BCH code and a subfield subcode of algebraic geometric code on a hyper elliptic curve are discussed as special cases.

In this paper, we show that the conventional BCH codes can be better than the AG codes when the number of check symbols is relatively small. More precisely, we consider an AG code on Cab whose number of check symbols is less than min {g+a, n-g}, where n and g denote the code length and the genus of the curve, respectively. It is shown that there always exists an extended BCH code, (i) which has the same designed distance as the Feng-Rao designed distance of the AG code and the code length and the rate greater than those of the AG code, or (ii) which has the same number of check symbols as that of the AG code, the designed distance not less than that of the AG code and the code length longer than that of the AG code.