## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

New combinatorial construction techniques are proposed which
convert binary user information into a (0,k) constrained sequence having
the virtue that at most k `zeroes' between logical `ones' will occur. In
this way sequences are constructed which have a limited runlength. These
codes find application in optical and magnetic recording systems. The
new construction methods provide efficient, high rate codes with a low
complexity. The low complex combinatorial structure of the encoder and
the decoder ensure a very fast and efficient parallel conversion of
binary information to codewords and vice versa. Specifically, we present
the combinatorial structures to convert 16 data bits into a 17 bit
constrained sequence to obtain an optimum (0,4) code, a (0,6) code with
at most one byte error propagation, and a (0,6/6)-code, respectively.
Serious error propagation is avoided by using constrained codes with
several unconstrained positions, which are reserved to store the parity
bits of an error control code which protects the constrained codeword

To read the full-text of this research,

you can request a copy directly from the authors.

... The FRB algorithm compares favorably with enumerative [15],[[17],Chap.6],[19], and combinatorial [20],[60] encoding: two important existing methods to generate (0, k) sequences. Specifically, the FRB encoding/decoding is simpler than enumeration , while achieving (asymptotically) similarly high encoding rates. ...

... Details of a rate 16/17, k = 6 code based on this technique can be found in [[17], pp.102103. Some other combinatorial constructions for high-rate (0, k) codes are discussed in [60]. An alternative to combinatorial techniques is to use enumerative coding (see [[17], ...

... The iterative pre-processing ideas were also extended to build fixed-rate (0, G/I) codes in Chapter 7.6. Two important existing methods to generate (0, k) sequences use enumerative [[17],Chap.6],[19], and combinatorial [20],[60] encoding. The FRB encoding/decoding is simpler than enumeration, while achieving (asymptotically) similarly high encoding rates. ...

Run-Length-Limited (RLL) channels are found in digital recording systems like the Hard Disk Drive (HDD), Compact Disc (CD), and Digital Versatile Disc (DVD). This thesis presents novel encoding algorithms for RLL channels based on a simple technique called bit stuffing. First, two new capacity-achieving variable-rate code constructions are proposed for (d,k) constraints. The variable-rate encoding ideas are then extended to (0,G/I) and other RLL constraints. Since variable-rate codes are of limited practical value, the second half of this thesis focuses on fixed-rate codes. The fixed-rate bit stuff (FRB) algorithm is proposed for the design of simple, high-rate (0,k) codes. The key to achieving high encoding rates with the FRB algorithm lies in a novel, iterative pre-processing of the fixed-length input sequence prior to bit stuffing. Detailed rate analysis for the proposed FRB algorithm is presented, and upper and lower bounds on the asymptotic (in input block length) encoding rate are derived. Several system-level issues of the proposed FRB codes are addressed, and FRB code parameters required to design rate 100/101 and rate 200/201 (0,k) codes are tabulated. Finally, the proposed fixed-rate encoding is extended to (0,G/I) constraints. Ph.D. Committee Chair: McLaughlin, Steven; Committee Member: Barnwell, Thomas; Committee Member: Barry, John; Committee Member: Fekri, Faramarz; Committee Member: Tetali, Prasad

... This codeword is protected by a systematic enor control code. The e parity bits are inserted in the codeword in a specific manner [21], such that the resulting sequence I c H c ,p.l .c · 1;1 ,·'C lp' l c ·1~1 Fig. 3. Block diagram of CC-EC Scheme II, a two-step modulation and error control coding scheme. ...

... The construction of a ( 0, G I I) code [2], [22] is in general more difficult than is the construction of a (0, k) code, because both the global (0, G) constraints and the (0, I) constraints on the interleaved sequences of elements with odd and even indices have to be fulfilled. In [23], a combinatorial construction of a rate 8/9, (0, 4/4) code and a rate 8/9, (0, 3/6) code have been presented, and in [14], [21], and [24], a rate 16/17, (0, 6/6) code has been presented. ...

... Otherwise, the constraints on the odd and even subsequence are violat~d. As an example, consider the rate 16/17, (0, 6/6) code [21]. By inserting pairs of unconstrained po- Table look-up can be employed to convert the other eight-bit source words into ten-bit codewords. ...

... This translates to low hardware requirements for the encoder and the decoder. Another method with similar aims but for a different class of line codes is described in [50,51]. ...

... A 8 -b it long code-word has one bit inserted in the middle.The maximum run-length in this code word was two before the insertion and it becomes three afterwards.However, by correctly designing the line code, it is possible in most cases to incorporate the inserted bits within the maximum run-length. This can be achieved by designing the line code to have unconstrained bit positions where single bits can be inserted w ithout affecting the maximum run[50]. Such a code has the property th a t it can be encoded and decoded independently of the value of those bits.Furthermore, in certain error correcting and line code combinations, it is possi ble to leave some of the information symbols uncoded and to distribute them , together with the parity symbols, in available unconstrained bit positions of the run-length limited sequence. ...

Channel coding is an important consideration influencing the design of a communications system. In particular, error control coding is used to detect and/or correct errors and line coding to modify the characteristics of the transmitted signal to suit other constraints of the channel, such as restricted frequency response. This thesis explores aspects of channel coding for such constrained channels with emphasis given to error control coding. Specifically, the hrst chapter of this thesis presents a general overview of channel coding, presents the organisation of the thesis and details the main contributions. The second chapter gives an overview of the principles of error control coding and line coding and explains a few terms that are connnonly used in the remainder of the thesis. One kind of constrained channel investigated here is the binary asymmetric error channel, where error transitions from one to zero occur with different probability than from zero to one. Error correcting codes for this channel and their properties are investigated in the third chapter. The fourth chapter introduces disparity control coding, and proposes a new error control coding structure that satisfies disparity constraints for both binary asymmetric and symmetric error channels. Run length limited channels are the subject of the hfth chapter. A new coding structure is proposed that offers advantages in performance over the one conventionally used for error control in such channels. The sixth chapter introduces peak power constraints present in multi-carrier systems. Codes that can be used limit the peak to average power ratio of such systems are presented and the application of the coding structure of the fifth chapter is also discussed. The final chapter brings the thesis to a conclusion by summarising the main results and proposing areas where further work may be fruitful.

... Fortunately, a block encoder/decoder architecture with acceptable implementation complexity for the constraints proposed here can always be designed by well known enumerative techniques [5]. A particularly efficient architecture based on the combinatorial techniques of [6] for one of the considered constraints is presented in [9] and outlined in Section V. ...

... In practice, there is always intertrack interference (ITI), i.e., the read head picks up magnetization from an adjacent track. Therefore, the channel output is given by (6) where is the discrete-time impulse response of the head to the adjacent track, and is the sequence recorded on that track. We assume that the noise is white. ...

During the past few years, significant progress has been made in
defining high capacity constraints which prohibit specified differences
between constrained sequences, thus ensuring that the minimum distance
between them is larger than for the uncoded system. However, different
constraints which avoid the same prescribed set of differences may have
different capacities, and codes into such constraints may have different
complexity of encoder/decoder architecture and different performance on
more realistic channel models. These issues, which have to be considered
in application of distance enhancing codes, are discussed here. We
define several distance enhancing constraints which support design of
high rate codes. We also define weak constraints for which the minimum
distance between sequences may be the same as for the uncoded system but
the number of pairs of sequences at tile minimum distance is smaller.
These constraints support design of even higher rate codes. We discuss
the implementation issues of both types of constraints as well as their
performance on the ideal channel and channels with colored noise and
intertrack interference

... This imposes limitations on the block length and the achievable rate. The majority of codes have a rate 8/9, and only recently, codes with longer block lengths have been developed, either by using algorithms [4], [14], [15] or by interleaving a short block code with uncoded bits [15]–[17]. The latter methods increase the rate and limit error propagation, but the constraints of the resulting code are certainly not the best possible for the given rate. ...

We present advanced combinatorial techniques for constructing
maximum runlength-limited (RLL) block codes and maximum transition run
(MTR) codes. These codes find widespread application in recording
systems. The proposed techniques are used to construct a high-rate
multipurpose modulation code for recording systems. The code, a rate
16/17, (0,3,2,2) MTR code, that also fulfills (0,15,9,9) RLL constraints
is a high-rate distance-enhancing code with additional constraints for
improving timing and gain control. The encoder and decoder have a
particularly efficient architecture and allow an instantaneous
translation of 16-bit source words into 17-bit codewords and vice versa.
The code has been implemented in Lucent read-channel chips and has
excellent performance

... 2. Possible realization of combined guided scrambling and MRL coding, where the sequence selector anticipates post-processing with an MRL code. uncoded source symbols as described in [4], [5]. One can get an additional degree of freedom in the selection of the scrambled sequences by using a strategy where the interspersed bits are observed and the source symbols of the underlying constrained code are transformed only if the overall constraints are violated, or, interestingly, the encoder may select one of the two possible sequences based on the overall constraints. ...

Methods are developed to effectively combine guided scrambling and maximum run-length limited codes in order to impose a guaranteed maximum run-length constraint. It will be demonstrated that the combination of guided scrambling and a well-chosen maximum run-length limited code may offer a sound trade-off between overall code rate and performance in terms of the probability of violating the channel constraints.

... ) codes [25], [33], MTR( ) codes [26], and PRML( ) codes [27] . The MTR code design methodologies presented in this section are based on the use of the state transition diagrams for MTR constraints and the application of the look-ahead coding technique and violation detection combined with substitution. ...

... The high rate´¼ Ð Ö µ codes presented in [5], [6] are particularly suitable for the construction of higher rate codes because of their tight constraints. This is achieved by interspersing the codewords with uncoded source symbols as described in [4], [5]. One can get an additional degree of freedom in the selection of the scrambled sequences by using a strategy where the interspersed bits are observed and the source symbols of the underlying constrained code are transformed only if the overall constraints are violated, or, interestingly, the encoder may select one of the two possible sequences based on the overall constraints. ...

Guided scrambling generates for each possible source word a unique set of candidate codewords and selects the "best" word subject to certain given channel constraints. This is an effective technique to generate long and highly efficient codes that satisfy the given channel constraints with high probability. These codes are referred to as weak constrained codes, because the guided scrambling method cannot guarantee that all constraints are satisfied. One of the common constraints, the maximum run-length constraint, is of particular importance, because sequences that violate this constraint are likely to cause loss of timing. For this reason, methods are developed in conjunction with guided scrambling to impose a guaranteed maximum run-length constraint. The performance of combined guided scrambling and maximum run-length limited codes is analyzed. It will be demonstrated that the combination of guided scrambling and a well-chosen maximum run-length limited code may offer a sound trade-off between overall code rate and performance in terms of the probability of violating the channel constraints.

Time-varying encoders for constrained systems are introduced. The
approach generalizes the state-splitting (ACH) algorithm in a way that
yields encoders consisting of multiple phases, with encoding proceeding
cyclically from one phase to the next. The framework is useful for
design of high-rate codes with reduced decoder error propagation and
reduced complexity

The sequence replacement technique converts an input sequence into a constrained sequence in which a prescribed subsequence is forbidden to occur. Several coding algorithms are presented that use this technique for the construction of maximum run-length limited sequences. The proposed algorithms show how all forbidden subsequences can be successively or iteratively removed to obtain a constrained sequence and how special subsequences can be inserted at predefined positions in the constrained sequence to represent the indices of the positions where the forbidden subsequences were removed. Several modifications are presented to reduce the impact of transmission errors on the decoding operation, and schemes to provide error control are discussed as well. The proposed algorithms can be implemented efficiently, and the rates of the constructed codes are close to their theoretical maximum. As such, the proposed algorithms are of interest for storage systems and data networks.

When a block modulation code is concatenated with an error-correction code (ECC) in the standard way, the use of a modulation code with long block lengths results in error propagation. This article analyzes the performance of modified concatenation, which involves reversing the order of modulation and the ECC. This modified scheme reduces the error propagation, provides greater flexibility in the choice of parameters, and facilitates soft-decision decoding, with little or no loss in transmission rate. In particular, examples are presented which show how this technique can allow fewer interleaves per sector in hard disk drives, and permit the use of more sophisticated block modulation codes which are better suited to the channel

We construct uniquely decodable (UD) code pairs for the binary multiplying channel without feedback, using pairs of binary codes. By taking appropriate cosets of linear codes with many information sets for these binary codes, we obtain new rate pairs in the zero-error capacity region Z of this channel. In particular, the rate pair (log(3/2),log(3/2)) is in Z and yields the largest known sum of the rates of pairs in Z. As this rate pair can be achieved with LTD pairs with equal members, we have obtained an asymptotically optimal construction for the combinatorial concept of cancellative families of sets.

This paper addresses a new scheme by Vasic and Pedagani to combine low-density parity-check (LDPC) codes with run length limited (RLL) constraints. In this method, the RLL constraints are embedded into the LDPC codewords by deliberately flipping the bits of LDPC codewords that violate the RLL constraints. It is important to keep the number of flipped bits small in order to not overburden the LDPC decoder. In this paper, we introduce a method to control the number of flipped bits by using pseudorandom sequences. We present a new low-complexity iterative decoding and detection scheme to correct both the flipped bits and channel errors in a partial response channel. Analyses and simulation results show that the proposed method has good performance and reasonable complexity for (0,k) RLL constraints.

A 16/20 (0, 6) dc- and Nyquist-free trellis code for the partial
response class-I (PR1) and class-II (PR2) channels has been developed.
With a Lorentzian pulse, this code showed a coding gain of 1.2 dB for
the PR1 channel and 2.2 db for the PR2 channel at a user bit density of
3.0, compared with the conventional 16/17 (0, 6/6) run-length limited
code for the extended partial response class-IV (EPR-I) channel in
error-rate simulation. With a pulse with dc response, this code showed a
coding gain of 0.5 dB for the PR1 channel and 2.4 dB for the PR2 channel
at the user bit density, compared with the 16/17 code for the EPR4
channel

Constrained sequence codes are widely used to meet constraints imposed by digital storage and communication systems. This paper develops an algorithm for the construction of constrained codes that admit state-independent decoding. By partitioning the code into a group of alphabets, one for each state, a codebook is developed using this algorithm that will allow the code to be decoded at the receiver without the need for state information. Finally, we use this algorithm to construct DC-free runlength-limited (RLL) codes, and we present two highly efficient state-independent decodable DC-free RLL codes.

In this paper, a modified equalization target is proposed for the
high density magnetic recording channel. This target is a closer match
to the channel than the EEPR4 response and hence has better detection
performance. Based on the dominant error events for this target, a
parity-based coding scheme is also proposed to achieve a coding gain
with the modified target. The use of the parity code detects the
occurrence of the dominant error events while achieving a high code
rate. The detection system consists of a Viterbi detector matched to the
channel response and a post processor to handle the code constraints.
This system is shown to perform well compared to other proposed
detection systems through analysis and simulation on a Lorentzian based
channel model

We introduce the fixed-rate bit stuff (FRB) algorithm for efficiently encoding and decoding maximum-runlength-limited (MRL) sequences. Our approach is based on a simple, variable-rate technique called bit stuffing . Bit stuffing produces near-capacity achieving codes for a wide range of constraints, but encoding is variable-rate, which is unacceptable in most applications. In this work, we design near-capacity fixed-rate codes using a three-step procedure. The fixed-length input data block first undergoes iterative preprocessing, followed by variable-rate bit stuffing, and finally dummy-bit padding to a fixed output length. The iterative preprocessing is key to achieving high encoding rates. We discuss rate computation for the proposed FRB algorithm and show that the asymptotic (in input block length) encoding rate is close to the average rate of the variable-rate bit stuff code. Then, we proceed to explore the effect of decreasing/increasing the number of preprocessing iterations. Finally, we derive a lower bound on the encoding rate with finite-length input blocks and tabulate the parameters required to design FRB codes with rate close to 100/101 and 200/201.

Binary data is mapped into constrained sequences, called (d,k) sequences, where d represents the minimum and k represents the maximum number of 0's between any pair of consecutive 1's. A code with (0,3) d,k constraints can be constructed using various code-word lengths. This is illustrated for the desirable code-word length eight, where 8 binary bits are mapped into 9 bit cells; this mapping provides for a rate of 8/9. The encoder logic for the mapping is derived.

We present a systematic procedure for mapping data sequences into
codewords of a prefix-synchronized code (PS-code), as well as for
performing the inverse mapping. A PS-code, proposed by Gilbert (1960),
belongs to a subclass of comma-free codes and is useful to recover word
synchronization when errors have occurred in the stream of codewords. A
PS-code is defined as a set of codewords with the property that each
codeword has a known sequence as a prefix, followed by a coded data
sequence in which this prefix is not allowed to occur. The largest
PS-code among all PS-codes of the same code length is called a maximal
prefix-synchronized code (MPS-code). We develop an encoding and decoding
algorithm for Gilbert's MPS-code with a prefix of the form 11...10 and
extend the algorithm to the class PS-codes of which the prefix is
self-uncorrelated. The computational complexity of the entire mapping
process is proportional to the length of the codewords

When messages are transmitted as blocks of binary digits, means of locating the beginnings of blocks are provided to keep the receiver in synchronism with the transmitter. Ordinarily, one uses a special synchronizing symbol (which is really a third kind of digit, neither 0 nor 1) for this purpose. The Morse code letter space and the teletype start and stop pulses are examples. If a special synchronizing digit is not available, its function may be served by a short sequence of binary digits P which is placed as a prefix to each block. The other digits must then be constrained to keep the sequence P from appearing within a block. If blocks of N digits (including the prefix P ) are used, the prefix should be chosen to make large the number G(N) of different blocks which satisfy the constraints. Lengthening the prefix decreases the number of "message digits" which remain in the block but also relaxes the constraints. Thus, for each N , there corresponds some optimum length of prefix. For each prefix P , a generating function, a recurrence formula, and an asymptotic formula for large N are found for G(N) . Tables of G(N) are given for all prefixes of four digits or fewer. Among all prefixes P of a given length A , the one for which G(N) has the most rapid growth is P = 11 cdots 1 . However, for this choice of P , the table of values of G(N) starts with small values; 11 cdots 1 does not become the best A -digit prefix until N is very large. At these values of N , the (A + 1) - digit prefix 11 cdots 10 is still better. The tables suggest that, for any N , a best prefix can always be found in the form 11 cdots 10 , for suitable A . Taking P = 11 cdots 10 and A = [log_2 (N log_2 e)] it is shown that G(N) is roughly 0-
.35N^{-1} X2^N . This result is near optimal since no choice of P can make G(N) exceed N^{-1}2^N .

Many of the types of modulation codes designed for use in storage devices using magnetic recording are discussed. The codes are intended to minimize the negative effects of intersymbol interference. The channel model is first presented. The peak detection systems used in most commercial disk drives are described, as are the run length-limited (d,k) codes they use. Recently introduced recording channel technology based on sampling detection-partial-response (or PRML) is then considered. Several examples are given to illustrate that the introduction of partial response equalization, sampling detection, and digital signal processing has set the stage for the invention and application of advanced modulation and coding techniques in future storage products.< >

It is shown how to detect or correct synchronization slippage in cyclic codes by using a prefix that incorporates the Fibonacci encoding technique developed by Kautz. Comparison with other methods shows that in many cases it seems to be more efficient. Implementation is simple. It is also shown how a similar technique can be used in comma-free codes. The resulting codes can be constructed algorithmic, ally from arbitrary information digits, that is, no table lookup is necessary. These codes seem to be the most efficient among those that require no code table.

A new family of codes is described for representing serial binary data, subject to constraints on the maximum separation between successive changes in value (0 rightarrow 1, 1 rightarrow , or both), or between successive like digits ( 0 's, 1 's, or both). These codes have application to the recording or transmission of digital data without an accompanying clock. In such cases, the clock must be regenerated during reading (receiving, decoding), and its accuracy controlled directly from the data itself. The codes developed for this type of synchronization are shown to be optimal, and to require a very small amount of redundancy. Their encoders and decoders are not unreasonably complex, and they can be easily extended to include simple error detection or correction for almost the same additional cost as is required for arbitrary data.

Significant improvements in magnetic storage densities have been
made feasible by the application of partial-response signaling combined
with maximum-likelihood sequence estimation. To enhance the performance
of this technique when applied to the class-IV partial-response channel,
which is recognized as being appropriate to model the magnetic recording
channel, it is often required to bound the number of consecutive zeros
in the recorded data sequence and its odd and even subsequences. We
investigate block codes that satisfy such a constraint. In particular,
we look for a set of maximal number of fixed-length sequences such that
any pair of them can be concatenated without violating the constraint.
In many cases, depending on the constraint and the length of the
sequences, we determine such a set, and in the remaining cases, we
determine at most three candidates for it. These results are used to
study the best possible constrained block codes

Self-clocking five bit record playback system

- G Milligan