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On the Number of Encoder States of a Type of RLL Codes.
Abstract
The relationship between the number of encoder states and the probable size of certain runlength-limited (RLL) codes is derived analytically. By associating the number of encoder states with (generalized) Fibonacci numbers, the minimum number of encoder states is obtained, which maximizes the rate of the designed code, irrespective of the codeword length.
... Note that the above inequalities are equivalent to the approximate eigenvector equation [1], and they are necessary conditions for code construction. Following these criteria, and by using either computer search or analytical approaches proposed in [18], we can determine the optimum number of encoder states to maximize the rate of the PRC code. The corresponding code rate ...
... First of all, a new (1,18) constrained single-bit even PC code is designed. The rate 9/13 (1,18) code with 5 states (i.e. r = 5, r 1 = 3, r 2 = 2) FSM proposed in [7] is used as the NC code, since its rate is 3.85% higher than that of the rate 2/3 d = 1 codes used in BD and HD-DVD systems. ...
... r = 5, r 1 = 3, r 2 = 2) FSM proposed in [7] is used as the NC code, since its rate is 3.85% higher than that of the rate 2/3 d = 1 codes used in BD and HD-DVD systems. A new rate 12/19 (1,18) code with 5 states is designed as the PRC code, which requires only 1 channel bit per parity bit with respect to the rate 2/3 d = 1 codes. Table I (ii), which also shows that 12-bit user data words can be supported. ...
This paper proposes a general and systematic code design method to efficiently combine constrained codes with parity-check (PC) codes for optical recording. The proposed constrained PC code includes two component codes: the normal constrained (NC) code and the parity-related constrained (PRC) code. They are designed based on the same finite state machine (FSM). The rates of the designed codes are only a few tenths below the theoretical maximum. The PC constraint is defined by the generator matrix (or generator polynomial) of a linear binary PC code, which can detect any type of dominant error events or error event combinations of the system. Error propagation due to parity bits is avoided, since both component codes are protected by PCs. Two approaches are proposed to design the code in the non-return-to-zero-inverse (NRZI) format and the non-return-to-zero (NRZ) format, respectively. Designing the codes in NRZ format may reduce the number of parity bits required for error detection and simplify post-processing for error correction. Examples of several newly designed codes are illustrated. Simulation results with the Blu-Ray disc (BD) systems show that the new d = 1 constrained 4-bit PC code significantly outperforms the rate 2/3 code without parity, at both nominal density and high density.
... It is much harder to define coding schemas when the coding rate is close to the capacity of the given (d,k) channel. An interesting study on successful allocation of codewords to the encoder states to maximise the code rate is given in Cai and Immink's (2006). The study shows that the number of encoder states, for the certain RLL codes, can always be associated with generalised Fibonacci numbers. ...
Comprehensive (d,k) sequences study is presented, complemented with the design of a new, efficient, Run-Length Limited (RLL) code. The new code belongs to group of constrained coding schemas with a coding rate of R = 2/5 and with the minimum run length between two successive transitions equal to 4. Presented RLL (4, ∞) code uses channel capacity highly efficiently, with 98.7% and consequently it achieves a high-density rate of DR = 2.0. It is implying that two bits can be recorded, or transmitted with one transition. Coding techniques based on the presented constraints and the selected coding rate have better efficiency than many other currently used codes for high density optical recording and transmission.
This paper is about an interdisciplinary knowledge relating to computer science and solid state electronics. The normal FORMAT or DELET operation in computer can not verily eras the data on the hard disk. By using precision instruments, the data could be recognized and recovered. Based on the intensive study in characteristic of magnetic medium and the data encoding rules, this paper represents an effective theory to destruct data from hard disk without specialized equipment. And finally, it proposes a feasible solution for the data destruction software on Linux and Windows operating system.
A method and system for encoding a segment of user data words into a segment of code words so that both modulation constraints and a predetermined parity-check constraint are satisfied. Each segment of the user data is partitioned into several data words, and encoded separately by first and second types of component code, which are referred to as the normal constrained code and the parity-related constrained code, respectively. The parity-check constraint over the combined code word is achieved by concatenating the sequence of normal constrained code words with a specific parity-related constrained code word chosen from a candidate code word set. Both the component codes are finite-state constrained codes, which are designed to have rates close to the Shannon capacity. Furthermore, they are based on the same finite state machine (FSM), which enables them to be connected seamlessly, without violating the modulation constraints. Two preferred embodiments are provided to design a code in the non-return-to-zero inverted (NRZI) format and the non-return-to-zero (NRZ) format, respectively. Designing the codes in NRZ format may reduce the number of parity-check bits required for error detection and simplify error correction or post-processing. The parity-check constraint is defined by the parity-check polynomial or parity-check matrix of a systematic linear block code, which could detect any type of dominant error event as well as error event combinations of a given optical recording system. As a result, the information density of the system is improved.
Introduction Let H be a Hilbert space. We consider quantum channel [4] with a nite input alphabet f1; :::; ag and with pure signal states S i = j i >< i j; i = 1; :::; a. Compound channel of length n is in the n{tensor product of the space H , i.e. H n = H : : : H . An input block (codeword) u = (i 1 ; : : : ; i n ) ; i k 2 f1; : : : ag; for it means using the compound state u = i 1 : : : i n 2 H n and the corresponding density operator S u = j <F9.792
Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature. I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to
avoid introducing new errors in the Second Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink
Rotterdam, November 2004
A sufficient condition on concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is noted. The validity of its sufficient condition is shown by some numerical computations.
Runlength-limited (RLL) codes, generically designated as (d, k) RLL codes, have been widely and successfully applied in modern magnetic and optical recording systems. The design of codes for optical recording is essentially the design of combined dc-free and runlength limited (DCRLL) codes. We will discuss the development of very efficient DCRLL codes, which can be used in upcoming generations of high-density optical recording products.
Ideas which have origins in C. E. Shannon's work in information theory have arisen independently in a mathematical discipline called symbolic dynamics. These ideas have been refined and developed in recent years to a point where they yield general algorithms for constructing practical coding schemes with engineering applications. In this work we prove an extension of a coding theorem of B. Marcus and trace a line of mathematics from abstract topological dynamics to concrete logic network diagrams.
In information theory, the reliability function and its bounds,
describing the exponential behavior of the error probability, are
important quantitative characteristics of the channel performance. From
a more general point of view, these bounds provide certain measures of
distinguishability of a given set of classical states. In a previous
paper, quantum analogs of the random coding and expurgation lower bounds
for the case of pure signal states were introduced. Here we discuss the
case of general quantum states, in particular, we prove the expurgation
bound conjectured previously and find the quantum cutoff rate for
arbitrary mixed signal states
It is shown that the capacity of a classical-quantum channel with
arbitrary (possibly mixed) states equals the maximum of the entropy
bound with respect to all a priori distributions. This completes the
recent result of Hausladen, Jozsa, Schumacher, Westmoreland, and
Wootters (1996), who proved the equality for the pure state
channel
We establish what we consider to be the definitive versions of Jensen's operator inequality and Jensen's trace inequality for functions defined on an interval. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis, and how Jensen's trace inequlity generalizes to C*-algebras..
Using random coding methods, lower estimates for the reliability function of a quantum channel with pure states are obtained. As a consequence, these estimates yield an alternative proof of a formula for the capacity of the channel. At zero rate, the precise value of the reliability function is found.
We have developed a new error correction method (Picket: a combination of a long distance code (LDC) and a burst indicator subcode (BIS)), a new channel modulation scheme (17PP, or (1, 7) RLL parity preserve (PP)-prohibit repeated minimum transition runlength (RMTR) in full), and a new address format (zoned constant angular velocity (ZCAV) with headers and wobble, and practically constant linear density) for a digital video recording system (DVR) using a phase change disc with 9.2 GB capacity with the use of a red (lambda=650 nm) laser and an objective lens with a numerical aperture (\mathit{NA}) of 0.85 in combination with a thin cover layer. Despite its high density, this new format is highly reliable and efficient. When extended for use with blue-violet (lambda≈ 405 nm) diode lasers, the format is well suited to be the basis of a third-generation optical recording system with over 22 GB capacity on a single layer of a 12-cm-diameter disc.
We study several inequalities for norms on matrices, in particular for the Hilbert–Schmidt and operator norms. These inequalities occur when comparing norms of the products XY and YX for matrices X and Y with suitable assumptions. we also point out some trace inequalities.
This paper describes our two newly developed high-density recording systems, which consist of a 35 GB system and a 20 GB system using partial response maximum likelihood (PRML) with blue laser diode (LD), and their respective evaluation systems. The 20 GB system consists of a 0.6 mm substrate disk system (System #1) with a numerical aperture (NA) 0.65 objective lens. On the other hand, the 35 GB system consists of a 0.1 mm cover layer disk system (System #2) with a NA0.85 objective lens. Moreover, we considered the two advantages and disadvantages, and future technical progress. Consequently, we considered that System #1 is fit for personal computer (PC) applications, and System #2 is fit for video disk recorder applications, such as prolonged high definition (HD) video recording of broadcasting satellite (BS) digital broadcasting.
Bell System Technical Journal, also pp. 623-656 (October)
Half-title pageSeries pageTitle pageCopyright pageDedicationPrefaceAcknowledgementsContentsList of figuresHalf-title pageIndex
Information theory answers two fundamental questions in communication theory: what is the ultimate data compression (answer: the entropy H), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We will argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam's Razor: “The simplest explanation is best”) and to probability and statistics (error rates for optimal hypothesis testing and estimation). The relationship of information theory to other fields is discussed. Information theory intersects physics (statistical mechanics), mathematics (probability theory), electrical engineering (communication theory) and computer science (algorithmic complexity). We describe these areas of intersection in detail.
We introduce a skew information of Lieb’s typefor selfadjoint matrices A, X. We give conditions for f and g so that Sf,g is positive or negative. As another important application, we settle the problem posed by Yanagi, Furuichi and Kuriyama from quantum information theory.
Certain trace inequalities related to matrix logarithm are shown. These results enable us to give a partial answer of the open problem conjectured by A.S. Holevo. That is, concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven in the case of 0 ⩽ s ⩽ 1.
We report on new runlength-limited codes (RLL) intended for the next generation of DVD. The efficiency of the newly developed RLL schemes is extremely close to the theoretical maximum, and as a result, significant density gains can be obtained with respect to prior art coding methods
A rate 16/17 maximum transition run (MTR) (3;11) code on an EEPRML
channel with a postprocessor (turbo-EEPRML) has been developed. The
postprocessor compensates for the degradation in performance due to
noise correlation. The proposed method has high performance and simple
circuitry. Simulation results showed that the rate 16/17 MTR (3;11) code
on a turbo-EEPRML channel has a performance gain of 1.0-4.0 dB over a
conventional extended partial response maximum likelihood channel with
the rate 16/17 run-length-limited code
A lower bound on the probability of decoding error of quantum communication channel is presented. The strong converse to the quantum channel coding theorem is shown immediately from the lower bound. It is the same as Arimoto's method except for the diculty due to noncommutativity. Keywords Quantum channel coding theorem, average error probability, strong converse, operator monotone 1 Introduction Recently, the quantum channel coding theorem was established by Holevo [9] and by Schumacher and Westmoreland [15], after the breakthrough of Hausladen et al. [7]. Furthermore, a upper bound on the probability of decoding error, in case rate below capacity, was derived by Burnashev and Holevo [2]. It is limited in pure signal state. They conjectured on a upper bound in general signal state, which corresponds to Gallager's bound [5] in classical information theory. We will show a lower bound on the probability of decoding error, in case rate above capacity, which corresponds to Arimoto's bou...
Codes for Mass Data Storage Systems The Netherlands: Shannon Foundation
- K A S Immink
K. A. S. Immink, Codes for Mass Data Storage Systems. Den Haag, The Netherlands: Shannon Foundation, 1999.
3016 DK Rotterdam, The Nether-lands (e-mail: immink@iem.uni-due.de; kees@immink.nl)
- Essen
- Turing Germany
- Inc
Essen, Germany, and Turing Machines Inc., 3016 DK Rotterdam, The Nether-lands (e-mail: immink@iem.uni-due.de; kees@immink.nl).
Theory and Applications 93-2218-E-194-012, NSC 94-2213-E-194-013, and NSC 94-2213-E-194-019. The author is with the Department of Communication Engineering
- S Vajda
- Lucas Numbers Fibonacci
- Golden Section
S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications. New York: Halsted, 1989. 93-2218-E-194-012, NSC 94-2213-E-194-013, and NSC 94-2213-E-194-019. The author is with the Department of Communication Engineering, National Chung-Cheng University, Min-Hsiung, Chia-Yi, 621 Taiwan, R.O.C.(e-mail: francis@ccu.edu.tw).
High-density recording systems using paritial response maximum likelihood with blue lader diode
- T Iwanaga
- S Ogkubo
- M Nakano
- M Kubota
- H Honma
- T Ide
T. Iwanaga, S. Ogkubo, M. Nakano, M. Kubota, H. Honma, T. Ide., and
R. Katayama, "High-density recording systems using paritial response
maximum likelihood with blue lader diode," Japan J. Appl. Phys., vol.
42, no. 2B, pt. 1, pp. 1042-1043, Feb. 2003.
Codes for Mass Data Storage Systems. Den Haag, The Netherlands: Shannon Foundation
- K A S Immink
K. A. S. Immink, Codes for Mass Data Storage Systems. Den Haag,
The Netherlands: Shannon Foundation, 1999.
- S Vajda
- Lucas Numbers
- The Golden
- Section
S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section:
Theory and Applications. New York: Halsted, 1989.