Construction algorithm for network error-correcting codes attaining the Singleton bound

We give a centralized deterministic algorithm for constructing linear network error-correcting codes that attain the Singleton bound of network error-correcting codes. The proposed algorithm is based on the algorithm by Jaggi et al. We give estimates on the time complexity and the required symbol size of the proposed algorithm. We also estimate the probability of a random choice of local encoding vectors by all intermediate nodes giving a network error-correcting codes attaining the Singleton bound. We also clarify the relationship between the robust network coding and the network error-correcting codes with known locations of errors.Comment: To appear in IEICE Trans. Fundamentals (http://ietfec.oxfordjournals.org/), vol. E90-A, no. 9, Sept. 2007. LaTeX2e, 7 pages, using ieice.cls and pstricks.sty. Version 4 adds randomized construction of network error-correcting codes, comparisons of the proposed methods to the existing methods, additional explanations in the proo

Algebraic geometric construction of a quantum stabilizer code

The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a construction method of such a self-orthogonal space using an algebraic curve. By using the proposed method we construct an asymptotically good sequence of binary stabilizer codes. As a byproduct we improve the Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this paper can be understood without knowledge of quantum mechanics.Comment: LaTeX2e, 12 pages, 1 color figure. A decoding method was added and several typographical errors were corrected in version 2. The description of the decoding problem was completely wrong in version 1. In version 1 and 2, there was a critical miscalculation in the estimation of parameters of codes, and the constructed sequence of codes turned out to be worse than existing ones. The asymptotically best sequence of quantum codes was added in version 3. Section 3.2 appeared in IEEE Transactions on Information Theory, vol. 48, no. 7, pp. 2122-2124, July 200

Improved Asymptotic Key Rate of the B92 Protocol

We analyze the asymptotic key rate of the single photon B92 protocol by using Renner's security analysis given in 2005. The new analysis shows that the B92 protocol can securely generate key at 6.5% depolarizing rate, while the previous analyses cannot guarantee the secure key generation at 4.2% depolarizing rate.Comment: IEEEtran.sty, 3 pages, 1 figure. Submitted to IEEE ISIT 201

Fidelity of a t-error correcting quantum code with more than t errors

It is important to study the behavior of a t-error correcting quantum code when the number of errors is greater than t, because it is likely that there are also small errors besides t large correctable errors. We give a lower bound for the fidelity of a t-error correcting stabilizer code over a general memoryless channel, allowing more than t errors. We also show that the fidelity can be made arbitrary close to 1 by increasing the code length.Comment: 9 pages, ReVTeX4 beta 5. To be published in Phys. Rev. A. The lower bound is made tighter in version 4 and 5. All approximations in the first version are removed, and a lower bound for the average of the fidelity is given in the second version. A critical error is correcte

Message Randomization and Strong Security in Quantum Stabilizer-Based Secret Sharing for Classical Secrets

We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert-Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explicit construction of strongly secure ramp secret sharing scheme by quantum stabilizers, which can support twice as large classical secrets as the McEliece-Sarwate strongly secure ramp secret sharing scheme of the same share size and the access structure.Comment: Publisher's Open Access PDF. arXiv admin note: text overlap with arXiv:1811.0521

Quantum Stabilizer Codes Can Realize Access Structures Impossible by Classical Secret Sharing

We show a simple example of a secret sharing scheme encoding classical secret to quantum shares that can realize an access structure impossible by classical information processing with limitation on the size of each share. The example is based on quantum stabilizer codes.Comment: LaTeX2e, 5 pages, no figure. Comments from readers are welcom