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 storage-effective enumerative encoding scheme is proposed for translating user data into long dk sequences and vice versa. For dk runlength-limited 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 Terms---Constrained code, enumerative coding, recording code, runlength-limited. I.

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a non-negative 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 order-preserving. 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 order-preserving 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

We design a succinct full-text index based on the idea of Huffman-compressing the text and then applying the Burrows-Wheeler transform over it. The resulting structure can be searched as an FM-index, 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.

New methods are presented to protect maximum runlength-limited 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 high-rate constrained codes, as error control can be provided with similar capabilities as for unconstrained sequences. Index Terms---Burst correction codes, enumerative coding, forward error correction, modulation coding, Reed--Solomon codes, runlength codes, synchronization. I.

When block modulation codes are concatenated with an error-correction code (ECC) in the standard way, the use of long block lengths results in error-propagation. 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 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 sophisticated block modulation codes which are better suited to the channel. Index terms: concatenated codes, Reed-Solomon codes, modulation codes, magnetic data storage 1 Introduction This paper is concerned with the interaction between the modulation code and the error-correcting code (ECC). The idea of modulation is to ensure that the sequence of bits transm...

this paper will concentrate, accepts the bit stream (extra bits added by the error-correction 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]

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 variable-length 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 "real-life" 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...

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.

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 block-decodable 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) runlength-limited code. The code is block-decodable, and it is particularly attractive as many commercially available Reed-Solomon codes operate in GF (2’). The encoder can be implemented with a simple 6-bit ROM, and decoding can be accomplished with a logic array.

This paper proposes and analyzes data synchronization techniques that not only resynchronize after encoded bits are corrupted by insertion, deletion or substitution errors, but also produce estimates of the time indices of the decoded data symbols, in order to determine their positions in the original source sequence. The techniques are based on block codes, and the estimates are of the time indices modulo some integer T , called the timing span, which is desired to be large. Several types of block codes that encode binary data are analyzed on the basis of the maximum attainable timing span for a given coding rate R (or, equivalently, redundancy ae = 1 \Gamma R) and permissible resynchronization delay D. It is found that relatively simple codes can asymptotically attain the maximum timing span among such block codes, which grows exponentially with delay, with exponent D(1 \Gamma R) + o(D). Thus large timing span can be attained with little redundancy and only moderate values of delay.