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27
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 35 (4 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
A Practical Method for Approaching the Channel Capacity of Constrained Channels
, 1997
"... A new coding technique is proposed that translates user information into a constrained sequence using very long codewords. Huge error propagation resulting from the use of long codewords is avoided by reversing the conventional hierarchy of the error control code and the constrained code. The new te ..."
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Cited by 34 (9 self)
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A new coding technique is proposed that translates user information into a constrained sequence using very long codewords. Huge error propagation resulting from the use of long codewords is avoided by reversing the conventional hierarchy of the error control code and the constrained code. The new technique is exemplified by focusing on (d; k)constrained codes. A storageeffective enumerative encoding scheme is proposed for translating user data into long dk sequences and vice versa. For dk runlengthlimited codes, estimates are given of the relationship between coding efficiency versus encoder and decoder complexity. We will show that for most common d; k values, a code rate of less than 0.5% below channel capacity can be obtained by using hardware mainly consisting of a ROM lookup table of size 1 kbyte. For selected values of d and k, the size of the lookup table is much smaller. The paper is concluded by an illustrative numerical example of a rate 256=466, (d =2,k= 15) code, which provides a serviceable 10% increase in rate with respect to its traditional rate 1=2, (2; 7) counterpart. Index TermsConstrained code, enumerative coding, recording code, runlengthlimited. I.
Maximum RunlengthLimited Codes with Error Control Capabilities
 IEEE J. Select. Areas Commun
, 2001
"... New methods are presented to protect maximum runlengthlimited sequences against random and burst errors and to avoid error propagation. The methods employ parallel conversion techniques and enumerative coding algorithms that transform binary user information into constrained codewords. The new sche ..."
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Cited by 20 (5 self)
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New methods are presented to protect maximum runlengthlimited sequences against random and burst errors and to avoid error propagation. The methods employ parallel conversion techniques and enumerative coding algorithms that transform binary user information into constrained codewords. The new schemes have a low complexity and are very efficient. The approach can be used for modulation coding in recording systems and for synchronization and line coding in communication systems. The schemes enable the usage of highrate constrained codes, as error control can be provided with similar capabilities as for unconstrained sequences. Index TermsBurst correction codes, enumerative coding, forward error correction, modulation coding, ReedSolomon codes, runlength codes, synchronization. I.
A Simple AlphabetIndependent FMIndex
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structu ..."
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Cited by 14 (7 self)
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We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structure. On a text of length n with zeroorder entropy H0, our index needs O(n(H0 + 1)) bits of space, without any significant dependence on σ. The average search time for a pattern of length m is O(m(H0 + 1)), under reasonable assumptions. Each position of a text occurrence can be located in worst case time O((H0 + 1)log n), while any text substring of length L can be retrieved in O((H0 + 1)L) average time in addition to the previous worst case time. Our index provides a relevant space/time tradeoff between existing succinct data structures, with the additional interest of being easy to implement. We also explore other coding variants alternative to Huffman and exploit their synchronization properties. Our experimental results on various types of texts show that our indexes are highly competitive in the space/time tradeoff map.
A Modified Concatenated Coding Scheme, with Applications to Magnetic Data Storage
 IEEE Trans. Inform. Theory
, 1997
"... When block modulation codes are concatenated with an errorcorrection code (ECC) in the standard way, the use of long block lengths results in errorpropagation. This paper analyzes the performance of modified concatenation, which involves reversing the order of modulation and ECC. This modified ..."
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Cited by 13 (3 self)
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When block modulation codes are concatenated with an errorcorrection code (ECC) in the standard way, the use of long block lengths results in errorpropagation. This paper analyzes the performance of modified concatenation, which involves reversing the order of modulation and ECC. This modified scheme reduces error propagation, provides greater flexibility in the choice of parameters, and facilitates softdecision 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 sophisticated block modulation codes which are better suited to the channel. Index terms: concatenated codes, ReedSolomon codes, modulation codes, magnetic data storage 1 Introduction This paper is concerned with the interaction between the modulation code and the errorcorrecting code (ECC). The idea of modulation is to ensure that the sequence of bits transm...
RunlengthLimited Sequences
 Proc. IEEE
, 1990
"... this paper will concentrate, accepts the bit stream (extra bits added by the errorcorrection system included) as its input and converts the stream to a waveform suitable for the specific recorder requirements. The object of the recording code is to bring structure into a data stream that is genera ..."
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Cited by 11 (1 self)
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this paper will concentrate, accepts the bit stream (extra bits added by the errorcorrection system included) as its input and converts the stream to a waveform suitable for the specific recorder requirements. The object of the recording code is to bring structure into a data stream that is generally not present in the information supplied by the user. All the aforementioned coding stages are present in a modern dig ital video recorder [1]
Robust Universal Complete Codes for Transmission and Compression
 Discrete Applied Mathematics
, 1996
"... Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is ..."
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Cited by 10 (4 self)
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Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and need not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several "reallife" examples. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...
Combinatorial Construction of High Rate Runlengthlimited Codes
 IEEE GLOBAL TELECOMMUN. CONF. GLOBECOM’96
, 1996
"... 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 applicat ..."
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Cited by 7 (4 self)
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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 code words 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 code word.
Encoding of dklrsequences using one weight set
 IEEE Trans. Inform. Theory
, 1996
"... The decoding function can be accomplished with a simple logic array. Note that the code above can easily be transformed into a (d = 1, k = 11) RLL code by representing the source word “2 ” by “000000 ” or “111111. ” If Q, ~6 = 00, then represent the source word “2 ” by “000000. ” It should be apprec ..."
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Cited by 5 (3 self)
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The decoding function can be accomplished with a simple logic array. Note that the code above can easily be transformed into a (d = 1, k = 11) RLL code by representing the source word “2 ” by “000000 ” or “111111. ” If Q, ~6 = 00, then represent the source word “2 ” by “000000. ” It should be appreciated that the smallest blockdecodable conventional rate 2/3 (d = 1, k = 11) code requires a codeword length of n = 18. This clearly shows that a design of an RLL code which is not a (d, k) code plus precoder can be quite advantageous. IV. CONCLUSIONS We have presented a new rate 4/6 (d = 1, k = 11) runlengthlimited code. The code is blockdecodable, and it is particularly attractive as many commercially available ReedSolomon codes operate in GF (2’). The encoder can be implemented with a simple 6bit ROM, and decoding can be accomplished with a logic array.
Combinatorial Representation of Generalized Fibonacci Numbers
, 1991
"... New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems. ..."
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Cited by 4 (0 self)
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New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems.