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Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require parity-check matrices to be in specific form may not fully be exploitable. Another issue of the recently proposed scheme is that the error correction capabilities for bit errors and phase errors are generally equal, which is not ideal for asymmetric error models. This paper addresses these two problems. We generalize the framework in such a way that any parity-check matrix of any binary or quaternary linear code can be exploited. Our generalization also allows for importing a pair of distinct linear codes so that error correction capabilities become suitably asymmetric., 8 pages, 1 figure

We present a nonintrusive method for reliably estimating the noise level during quantum computation and quantum communication protected by quantum error-correcting codes. As preprocessing of quantum error correction, our scheme estimates the current noise level through a negligible amount of classical computation using error syndromes and updates the decoder's knowledge on the spot before inferring the locations of errors. This preprocessing requires no additional quantum interaction or modification in the system. The estimate can be of higher quality than the maximum likelihood estimate based on perfect knowledge of channel parameters, thereby eliminating the need of the unrealistic assumption that the decoder accurately knows channel parameters a priori. Simulations demonstrate that not only can the decoder pick up on a change of channel parameters, but even if the channel stays the same, a quantum low-density parity-check code can perform better when the decoder exploits the on-the-spot estimates instead of the true error probabilities of the quantum channel., Comment: 5 pages, 1 figure

A Steiner 2-design of block size k is an ordered pair (V, B) of finite sets such that B is a family of k-subsets of V in which each pair of elements of V appears exactly once. A Steiner 2-design is said to be r-even-free if for every positive integer i =< r it contains no set of i elements of B in which each element of V appears exactly even times. We study the even-freeness of a Steiner 2-design when the cyclic group acts regularly on V. We prove the existence of infinitely many nontrivial Steiner 2-designs of large block size which have the cyclic automorphisms and higher even-freeness than the trivial lower bound but are not the points and lines of projective geometry., 12 pages, no figures

An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd��s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every admissible order v there exists a strictly 4-sparse 4CS(v). We also prove that for any positive integer r > 1 and sufficiently large integer v there exists a constant number c such that there exists a strictly r-sparse 4-cycle packing of order v with cv^2 4-cycles., 11 pages, no figures