For a given curve X and divisor class C, we give lower bounds on the degree of a divisor A such that A and A − C belong to specified semigroups of divisors. For suitable choices of the semigroups we obtain (1) lower bounds for the size of a party A that can recover the secret in an algebraic geometric linear secret sharing scheme with adversary threshold C, and (2) lower bounds for the support A of a codeword in a geometric Goppa code with designed minimum support C. Our bounds include and improve both the order bound and the floor bound. The bounds are illustrated for two-point codes on general

a b s t r a c tVarious methods have been used to obtain improvements of the Goppa lower bound for the minimum distance of an algebraic geometric code. The main methods divide into two categories, and all but a few of the known bounds are special cases of either the Lundell-McCullough floor bound or the Beelen order bound. The exceptions are recent improvements of the floor bound by Güneri, Stichtenoth, and Taskin, and by Duursma and Park, and of the order bound by Duursma and Park, and by Duursma and Kirov. In this paper, we provide short proofs for all floor bounds and most order bounds in the setting of the van Lint and Wilson AB method. Moreover, we formulate unifying theorems for order bounds and formulate the DP and DK order bounds as natural but different generalizations of the Feng-Rao bound for one-point codes.

We prove a formula for the minimum distance of two-point codes on a Hermitian curve.

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To investigate the current quality control status of raw materials in "Ssanghwatang", a widely used oriental formula effective for fatigue recovery in Korea, 1024 samples of 9 herbs available in Seoul area markets were collected, and the contents of quality inspection items such as total ash, acid-insoluble ash, loss on drying, and hazardous heavy metals (Pb, As, Cd, Hg) were examined. Medicinal herbs requiring quality control, which may not meet the specifications and standards set by the Korea Food and Drug Administration, were selected using robust z-scores, a random variable. Among the quality control items, control of total ash content was required for the Ssanghwatang raw materials Cinnamomi Cortex, Rehmanniae Radix Preparata, Zingiberis Rhizoma, and Zizyphi Fructus; loss on drying for Cinnamomi Cortex; and heavy metal (Cd) for Angelicae Gigantis Radix, Cinnamomi Cortex, and Zingiberis Rhizoma. Additionally, acid-insoluble ash content revealed how many fine soil and sand particles are present in commercial medicinal herbs, which were correlated with heavy metals such as Pb (r = 0.528) and As (r = 0.342) in Rehmanniae Radix Preparata (p < 0.01).