We show a construction of a quantum ramp secret sharing scheme from a nested pair of linear codes. Necessary and sufficient conditions for qualified sets and forbidden sets are given in terms of combinatorial properties of nested linear codes. An algebraic geometric construction for quantum secret sharing is also given.

We introduce a coding theoretic criterion for Yamamoto's strong security of the ramp secret sharing scheme. After that, by using it, we show the strong security of the strongly multiplicative ramp secret sharing proposed by Chen et al. in 2008.

We propose an encoding method for one-point algebraic geometry codes that usually requires less computation than the ordinary systematic encoder.

In this paper, we propose a lower bound for the minimum distance of [n,k] linear codes which are specified by generator matrices whose rows are k vectors of a given sequence of n linearly independent vectors over a finite field. The Feng-Rao bound and the order bound give the lower bounds for the minimum distance of the dual codes of the codes considered in this paper. We show that the proposed bound gives the true minimum distance for Reed-Solomon and Reed-Muller codes and exceeds the Goppa bound for some L-type algebraic geometry codes.

We show how to apply the Feng-Rao decoding algorithm and the Feng-Rao bound for the Ω-construction of algebraic geometry codes to the L-construction. Then we give examples in which the L-construction gives better linear codes than the Ω-construction in certain range of parameters on the same curve.

When we have a singular Cab curve with many rational points, we had better to construct linear codes on its normalization rather than the original curve. The only obstacle to construct linear codes on the normalization is finding a basis of L( Q) having pairwise distinct pole orders at Q, where Q is the unique place of the Cab curve at infinity. We present an algorithm finding such a basis from defining equations of the normalization of the original Cab curve.

This paper investigates the error correcting capabilities of concatenated codes employing algebraic geometry codes as outer codes and time-varying randomly selected inner codes, used on discrete memoryless channels with maximum likelihood decoding. It is proved that Gallager's random coding error exponent can be obtained for all rates by such codes. Further, it is clarified that the error exponent arbitrarily close to Gallager's can be obtained for almost all random selections of inner codes with a properly chosen code length, provided that the length of the outer code is sufficiently large. For a class of regular channels, the result is also valid for linear concatenated codes, and Gallager's expurgated error exponent can be asymptotically obtained for all rates.