We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting code
s from ternary outer codes. We show that some of the Varshamov-Tenengolts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. In the binary case, Varshamov has shown that almost all good linear codes for the asymmetric channel are also good for the symmetric channel. Our results indicate that Varshamovs argument does not extend to the nonbinary case, i.e., one can find better linear codes for asymmetric channels than for symmetric ones.
We introduce the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct fa
milies of new single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any stabilizer codes for finite block length, but also asymptotically achieve the quantum Hamming bound for large block length.
