By choosing specific divisors, we show that some algebraic-geometry (AG) codes from the Hermitian curves have better parameters than one-point Hermitian codes. The general lower bound is discussed.

It is shown that quantum error correction can be achieved by the use of classical binary codes or additive codes over F/sub 4/. In this correspondence, with the help of some algebraic techniques the theory of algebraic-geometric codes is used to construct an asymptotically good family of quantum error-correcting codes and other classes of good quantum error-correcting codes. Our results are compared with the known best quantum codes.

It is known that quantum error correction can be achieved using classical binary codes or additive codes over F/sub 4/. Asymptotically good quantum codes have been constructed from algebraic-geometry codes and a bound on (/spl delta/, R) was computed from the Tsfasman-Vladut-Zink bound of the theory of classical algebraic-geometry codes. In this correspondence, by the use of a concatenation technique we construct a family of asymptotically good quantum codes exceeding the bound in a small interval.

Ghorpade and Lachaud (1998) introduced Schubert codes as codes corresponding to the protective system (Tsfasman and Vladut 1991) defined by Schubert subvarieties of Grassmann varieties Gr(l,m). They gave an upper bound for the minimum distances of these codes in Ghorpade and Lachaud and conjectured their true minimum distances. In this note we prove the conjecture in the case of l=2.

We give codes on all members after level 4 of the family of Garcia-Stichtenoth (see Invent. Math., vol.121, p.211-22, 1995) curves with the property d/sub true/>d/sub FR/. These codes also give an improvement on Tsfasman-Vladut-Zink bound.

In this correspondence we give geometric codes on one of the curves of Garcia-Stichtenoth (1995) with the property that the true distances are bigger than Feng-Rao distances. Moreover, an example for which the Feng-Rao algorithm cannot correct more than [(d/sub FR/-1)/2]+1(<[(d/sub true/-1)/2]) errors is given.

Motivated by the paper of Calderbank, McGuire, Kumar, and Helleseth (see ibid., vol.42, no.1, p.217-26, Jan. 1996) we prove the following result: for any given positive integer l/spl ges/3, the minimum Lee weights of Hensel lifts (to Z/sub 4/) of binary primitive BCH codes of length 2/sup m/-1 and designed distance 2/sup l/-1 is just 2/sup l/-1 when (a) m can be divided by l or (b) m is sufficiently large. For Hensel lifts of binary primitive BCH codes of arbitrary designed distance /spl delta//spl ges/4, we also prove that their minimum Lee weight d/sub L//spl les/2([log/sub 2//spl delta/]+1)-1 when m is sufficiently large. Moreover, a result about minimum Lee weights of certain Z/sub 4/ codes defined by Galois rings, which is similar to the result in Calderbank et al., is proved.

In coding theory, it is of great interest to know the maximal length of MDS codes. In fact, the main conjecture says that the length of MDS codes over F/sub q/ is less than or equal to q+1 (except for some special cases). Munuera (see ibid., vol.38, p.1573-7, 1992) proposed a new way to attack the main conjecture on MDS codes for geometric codes. In particular, he proved the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough. He also asked whether a similar theorem can be proved for any hyperelliptic curve. The purpose of this correspondence is to give an affirmative answer. In fact, our method also proves the main conjecture for geometric MDS codes for q=2 if the genus of the hyperelliptic curve is either 1, 2 or 3, and for q=3 if the genus of the curve is 1.