Upper bounds on the `information leakage' of concatenated encoders for wiretap channels are presented. An illustrative implication of this result is that for a wide class of wiretap channels, there exists a sequence of encoder-decoder pairs such that the sequence achieves any rate below the secrecy capacity and each encoder in the sequence is constructible in polynomial time in its block length.

A polynomial-time construction of a sequence of self-orthogonal geometric Goppa codes attaining the Tsfasman-Vladut-Zink (TVZ) bound is presented. The issue of constructing such a code sequence was addressed in a context of constructing quantum error-correcting codes (Ashikhmin et al., 2001). Naturally, the obtained construction has implications on quantum error-correcting codes. In particular, the best known asymptotic lower bounds on the largest minimum distance of polynomially constructible quantum error-correcting codes are improved.

A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon-theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.

The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably

This paper describes the security of quantum key distribution (QKD) to share a random secret string of digits between two parties based on the principle of quantum mechanics. A Colderbank-Shor-Steane quantum code had been implicitly used in the BB84 protocol, which makes fidelity of this code to unity and the mutual information between the shared key and the data obtained approaches zero. In the BB84 protocol, the sender transmits digits from which encoded into an orthogonal basis or the conjugate basis. These CSS codes achieve a higher rate called as Shannon rate.

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.

The burst weight distributions (spectra) of linear maximum-Hamming-distance-separable (MDS) codes are all established explicitly.

It is shown that the set of deterministic random sequences (of symbols from a finite alphabet) with respect to a computable probability measure /spl mu/, in Martin-Lof's (1966) sense, and the set of deterministic random sequences with respect to another computable probability measure /spl nu/ are disjoint if /spl mu/ and /spl nu/ are different and the measures are either i.i.d. or homogeneous finite-order irreducible Markov measures.

A new class of error control codes, single byte error correcting and single byte plus single bit error detecting codes, are presented. The codes are suitable for semiconductor memory systems organized in a b-bit-per-chip manner, b/spl ges/2, and more efficient than previously known codes with as strong error control capabilities.