Before we gave a fast generalized minimum distance (GMD) decoding algorithm for one-point algebraic-geometry (AG) codes. In this paper, we propose another fast GMD decoding algorithm for these codes, where the present method includes an erasure deletion procedure while the past one uses an erasure addition procedure. Both methods find a minimal polynomial set of a given syndrome array, which is a candidate for an erasure-and-error locator polynomial set constrained with an erasure locator set of each size. Although both erasure addition and deletion GMD decoding algorithms have been established for one-dimensional algebraic codes such as RS codes, nothing but the erasure addition GMD decoding algorithm for multidimensional algebraic codes such as one-point AG codes have been given. The present erasure deletion GMD decoding algorithm is based on the Berlekamp-Massey-Sakata (BMS) algorithm from the standpoint of constrained multidimensional shift register synthesis. It is expected that both our past and present methods play a joint role in decoding for one-point AG codes up to the error correction bound.

A compound error is any combination of burst errors with various burst lengths including random errors. The compound weight of any such error is defined as a kind of combinational metric which is a generalization of Gabidulin's metric. First, we present a fast method for calculating the weight of any word. Based on this method, as an extension of Wadayama's augmenting method in the case of Hamming weight, we propose a method of constructing codes having higher coding rate by augmenting any compound-error-correcting codes. Furthermore, we show some examples of good compound-error-correcting codes obtained by using our augmenting method.

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.

In this paper we propose a method of constructing quasi-cyclic regular LDPC codes from a cyclic difference family, which is a kind of combinatorial design. The resulting codes have no 4-cycle, i.e. cycles of length four and are defined by a small set of generators of codes with high rate and large code length. In particular, for LDPC codes with column weight three, we clarify the conditions on which they have no 6-cycle and their minimum distances are improved. Finally, we show the performance of the proposed codes with high rates and moderate lengths.