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東京工芸大学工学部紀要

All Error Detecting Modified Hamming Codes for Ideal Optical Channel and Application to Practical System

Chugo Fujihashi

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

Modified Hamming Codes designed to have the capability of all error detecting and one error correcting on an ideal optical channel are presented. The codes are applicable to an optical channel with only quantum noise, and are able to achieve all error excluding transmission on the optical channel in a future advanced communication system. The codes are obtained from the Modified of ordinary (7, 4) Hamming codes by adding an extra one-bit symbol to make all errors detectable. It is shown that the codes are also useful on a practical level to completely exclude almost all errors on a channel affected by not only quantum noise but also external noise.

Related papers

All Error Detecting Modified Hamming Codes for Ideal Optical Channel and Application to Practical System

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All Error Detecting Modified Hamming Codes for Ideal Optical Channel and Application to Practical SystemChugo Fujihashi *1 Modified Hamming Codes designed to have the capability of all error detecting and one error correcting on an ideal optical channel are presented.The codes are applicable to an optical channel with only quantum noise, and are able to achieve all error excluding transmission on the optical channel in a future advanced communication system.The codes are obtained from the Modified of ordinary (7, 4) Hamming codes by adding an extra one-bit symbol to make all errors detectable.It is shown that the codes are also useful on a practical level to completely exclude almost all errors on a channel affected by not only quantum noise but also external noise.

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arXiv:quant-ph/9604006•Apr 8, 1996•71 citations
Modified Hamming Codes designed to have the capability of all error detecting and one error correcting on an ideal optical channel are presented. The codes are applicable to an optical channel with only quantum noise, and are able to achieve all error excluding transmission on the optical channel in a future advanced communication system. The codes are obtained from the Modified of ordinary (7, 4) Hamming codes by adding an extra one-bit symbol to make all errors detectable. It is shown that the codes are also useful on a practical level to completely exclude almost all errors on a channel affected by not only quantum noise but also external noise.
All Error Detecting Modified Hamming Codes for Ideal Optical Channel and Application to Practical SystemChugo Fujihashi *1 Modified Hamming Codes designed to have the capability of all error detecting and one error correcting on an ideal optical channel are presented.The codes are applicable to an optical channel with only quantum noise, and are able to achieve all error excluding transmission on the optical channel in a future advanced communication system.The codes are obtained from the Modified of ordinary (7, 4) Hamming codes by adding an extra one-bit symbol to make all errors detectable.It is shown that the codes are also useful on a practical level to completely exclude almost all errors on a channel affected by not only quantum noise but also external noise.
In the preceding chapter we mentioned the inevitable errors that occur when one tries to send quantum signals over, say, an optical fiber, even when there is no eavesdropper. But errors in transmission are not a problem just for quantum cryptography. For this entire chapter we forget about sending quantum information and instead focus on simply transmitting ordinary data faithfully over some kind of channel. Moreover, we assume that the data either is not sensitive or has already been encrypted. Unfortunately, many methods for transmitting data are susceptible to outside influences that can cause errors. How do we protect information from these errors? Error-correcting codes provide a mathematical method of not only detecting these errors, but also correcting them. Nowadays error-correcting codes are ubiquitous; they are used, for example, in cell-phone transmissions and satellite links, in the representation of music on a compact disk, and even in the bar codes in grocery stores. The story of modern error-correcting codes began with Claude Shannon's famous paper “A Mathematical Theory of Communication,” which was published in 1948. Shannon worked for Bell Labs where he specialized in finding solutions to problems that arose in telephone communication. Quite naturally, he started considering ways to correct errors that occurred when information was transmitted over phone lines. Richard Hamming, who also worked at Bell Labs on this problem, published a groundbreaking paper in 1950 on the subject.
Quantum computing is a computer development technology that uses quantum mechanics to perform the operations of data and information. It is an advanced technology, yet the quantum channel is used to transmit the quantum information which is sensitive to the environment interaction. Quantum error correction is a hybrid between quantum mechanics and the classical theory of error-correcting codes that are concerned with the fundamental problem of communication, and/or information storage, in the presence of noise. The interruption made by the interaction makes transmission error during the quantum channel qubit. Hence, a quantum error correction code is needed to protect the qubit from errors that can be caused by decoherence and other quantum noise. In this paper, the digital system design of the quantum error correction code is discussed. Three designs used qubit codes, and nine-qubit codes were explained. The systems were designed and configured for encoding and decoding nine-qubit error correction codes. For comparison, a modified circuit is also designed by adding Hadamard gates.
Quantum error-correcting codes so far proposed have not worked in the presence of noise which introduces more than one bit of entropy per qubit sent through a quantum channel, nor can any code which identifies the complete error syndrome. We describe a code which does not find the complete error syndrome and can be used for reliable transmission of quantum information through channels which add more than one bit of entropy per transmitted bit. In the case of the depolarizing channel our code can be used in a channel of fidelity .8096. The best existing code worked only down to .8107.