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Two constructions of a low-complexity near-optimal detection method of dc2-balanced codes are presented. The methods presented are improvements on Slepian's algorithm for optimal detection of permutation codes.

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... The number of dc 2 -balanced codewords [7], [10], N dc 2 (n), for asymptotically large n, n mod 4 = 0, can be found by substituting s = n 2 = µ s and p = n(n + 1) 4 = µ p , into (19). Then we obtain ...

We consider the transmission and storage of data that use coded binary symbols over a channel, where a Pearsondistance-based detector is used for achieving resilience against additive noise, unknown channel gain, and varying offset. We study Minimum Pearson Distance (MPD) detection in conjunction with a set, S, of codewords satisfying a center-of-mass constraint. We investigate the properties of the codewords in S, compute the size of S, and derive its redundancy for asymptotically large values of the codeword length n. The redundancy of S is approximately 3/2 log2 n + α where α = log2 √π/24 =-1.467. for n odd and α =-0.467. for n even. We describe a simple encoding algorithm whose redundancy equals 2 log2 n + o(log n). We also compute the word error rate of the MPD detector when the channel is corrupted with additive Gaussian noise.

Codes were designed for optical disk recording system and future options were explored. The designed code was a combination of dc-free and runlength limited (DCRLL) codes. The design increased minimum feature size for replication and sufficient rejection of low-frequency components enabling a simple noise free tracking. Error-burst correcting Reed-Solomon codes were suggested for the resolution of read error. The features of DCRLL and runlength limited (RLL) sequences was presented and practical codes were devised to satisfy the given channel constraints. The mechanism of RLL codes supressed the components of the genarated sequences. The construction and performance of alternative Eight to fourteen modulation (EFM)-like codes was studied.

An efficient algorithm is presented for encoding unconstrained
information sequences into a third-order spectral-null code of length n
and redundancy 9log<sub>2</sub> n+O(log log n). The encoding can be
implemented using O(n) integer additions and O(nlog n) counter
increments

Let A ( n ) be the number of partitions of { 1 , … , n } into two sets A, B of cardinality n /2 such that . Then there is the asymptotic result

The role of line coding is to convert source data to a digital form resistant to noise in combination with such other impairments as a specific medium may suffer (notably intersymbol interference, digit timing jitter and carrier phase error), while being reasonably economical in the use of bandwidth. This paper discusses the nature and role of various constraints on code words and word sequences, including those commonly used on metallic lines, optical fibres, carrier channels and radio links ; and gives some examples from each of these applications. It should serve both as a general review of the subject and as an introduction to the companion papers on specific topics.

Let T(n,k) denote the set of all words of length n over the alphabet {+1, -1}, having a kth order spectral-null at zero frequency. A subset of T(n,k) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of T(n,k) are derived. In particular we prove that (k - 1) log2 (n/k) less-than-or-equal-to n - log2 \T(n,k)\ less-than-or-equal-to O(2k log2 n) for infinitely many values of n. On the other hand, we show that T(n,k) is empty unless n is divisible by 2m, where m = left-perpendicularlog2 kright-perpendicular + 1. Furthermore, bounds on the minimum Hamming distance d of T(n,k) are provided, showing that 2k less-than-or-equal-to d less-than-or-equal-to k(k - 1) + 2 for infinitely many n. We also investigate the minimum number of sign changes in a word x is-an-element-of T(n,k) and provide an equivalent definition of T(n,k) in terms of the positions of these sign changes. An efficient algorithm for encoding arbitrary information sequences into a second-order spectral-null code of redundancy 3 log2 n + O(log log n) is presented. Furthermore, we prove that the first nonzero moment of any word in T(n,k) is divisible by k! and then show how to construct a word with a spectral null of order k whose first nonzero moment is any even multiple of k!. This leads to an encoding scheme for spectral-null codes of length n and any fixed order k, with rate approaching unity as n --> infinity.

An efficient recursive method has been proposed for the encoding/decoding of second-order spectral-null codes, via concatenation by Tallini and Bose. However, this method requires the appending of one, two, or three extra bits to the information word, in order to make a balanced code, with the length being a multiple of 4; this introduces redundancy. Here, we introduce a new quasi-second-order spectral-null code with the length equiv 2 (mod 4) and extend the recursive method of Tallini and Bose, to achieve a higher code rate

A method is presented for designing binary channel codes in such a way that both the power spectral density function and its low-order derivatives vanish at zero frequency. The performance of the new codes is compared with that of channel codes designed with a constraint on the unbalance Of the number of transmitted positive and negative pulses. Some remarks are made on the error-correcting capabilities of these codes.

A class of codes and decoders is described for transmitting digital information by means of bandlimited signals in the presence of additive white Gaussiau noise. The system, called permutation modulation, has many desirable features. Each code word requires the same energy for transmission. The receiver, which is maximum likelihood, is algebraic in nature, relatively easy to instrument, and does not require local generation of the possible sent messages. The probability of incorrect decoding is the same for each sent message. Certain of the permutation modulation codes are more efficient (in a sense described precisely) than any other known digital modulation scheme. PCM, ppm, orthogonal and biorthogonal codes are included as special cases of permutation modulation.

Higher‐order spectral‐null codes: constructions and bounds

- Roth R.M.