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# Spectrum shaping with binary dc^2 -constrained channel codes

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A method is presented for designing binary channel codes in such a way that both the power spectral density function and its second-derivative vanish at zero frequency. Recursion relations are derived to determine the number of codewords that can be used in this coding scheme. A simple algorithm for encoding and decoding codewords is developed. The performance of the new codes is compared with that of classical channel codes designed with a constraint on the unbalance of the number of transmitted positive and negative pulses.
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... However, this method adds complexity in encoding and decoding balanced codes, and the size of the codebook is relatively small. The generating function offers a tool for enumerating the balanced codes [44,45]. Encoding/decoding of balanced codes has attracted a considerable amount of research and engineering attention [46,47]. ...
... Next, we will use (3.23) as the decoding criterion in simulations of two codes. 44 3. NOISY CHANNELS WITH UNKNOWN OFFSET MISMATCH ...
... Let N sn 2 (n) denote the number of second-order spectral-null code codewords. It has been found in [44] that N sn 2 (n) = 0 if n mod 4 = 0, and for asymptotically large n the number of second-order spectral-null codewords equals N sn 2 (n) 4 3 π 2 n n 2 , n mod 4 = 0. (6.4) ...
... Let SN (n, q) indicate the set of qth-order spectral-null words in φ n , with φ = {−1, +1} a bipolar alphabet. This set is defined as (see [6], [16], [7]) ...
... C is a q-OSN(n,k) code of length n and withk information bits, if, and only if C ⊆ SN (n, q) and |C| = 2k. In the case q = 1, the q-OSN(n,k) codes coincide with the balanced codes [6], [8], [4], [2], [3], [16], [19], [21], [26], [32], [23], [10], [24], [25], [27], [13], [15], [14]. On the other hand, for q ≥ 2, the q-order spectral null codes are applied in digital recording and partial-response channels [7], [16]. ...
... S2: for the m 0 -balanced information word X = 010101110001, we can choose one of the following m 1balanced codewords as the encoding of X: E 2 (X) = X (0) C 0 = 110001100110 00110110 E 2 (X) = X (15) C 1 = 100011000111 11100100 E 2 (X) = X (53) C 5 = 011010100011 10101001 E 2 (X) = X (56) C 6 = 011011000011 10100101 E 2 (X) = X (57) C 7 = 011011000011 11000011 Note that, there is more than one balancing index for the information word X = 110001100110. This fact holds for all the information words in S (12,6). Therefore, we can exploit the freedom of choice of the balancing indices to convey extra auxiliary data, thereby reducing the overall redundancy. ...
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The code design problem of non-recursive second-Order Spectral Null (2-OSN) codes is to convert balanced information words into 2-OSN words employing the minimum possible redundancy. Let k be the balanced information word length. If k∈2IN then the 2-OSN coding scheme has length n = k +r, with 2-OSN redundancy r∈2IN and n∈4IN. Here, we use a scheme with r = 2 log k + Θ(log log k). The challenge is to reduce redundancy even further for any given k. The idea is to exploit the degree of freedom to select from more than one possible 2-OSN encoding of a given balanced information word. To reduce redundancy, empirical results suggest that extra information δk = 0:5 log k + Θ(log log k) is obtained. Thus, the proposed approach would give a smaller redundancy r’ = 1:5 log k + Θ(log log k) less than r = 2 log k + Θ(log log k).
... Define [7], [8] h n (x, y) = (1 + xy)(1 + xy 2 ) . . . (1 + xy n ). ...
... h n (x, y). we can write down a recursive relation for the coefficients of h n (x, y) [7], [8], where for clerical convenience we use the notation ...
... 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 ...
Article
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.
... In the prior art, constrained coding methods have been presented that offer a cure for the reported error performance degradation. For example, in [4], constrained codes are advocated, where the codebook S is chosen such that each codeword x ∈ S satisfies two conditions, namely ...
... Properties and constructions of dc 2 -balanced codes were first presented by Immink [4]. Efficient code constructions were presented by, for example, Yang [6], Tallini and Bose [7]. ...
... Let the number of dc 2 -balanced codewords of length n be denoted by N dc 2 (n). It has been found in [4] that N dc 2 (n) = 0 if n mod 4 ̸ = 0. The number of dc 2 -balanced codewords equals for asymptotically large n [8] (see also Section VI) ...
... Using the theory, Theorem 2, developed in [9], we may slightly reduce the computational effort, but its application also requires the evaluation of each codeword in S K . The modification of the enumeration method shown for K = 2 [15] is prohibitive for large n and K. ...
... For n ≤ 32 we applied an exhaustive search for computing (2) of S 2 . For n > 32, we applied a modified enumeration method introduced in [15]. For n = 16, we notice a raggedness of the correlation function,ρ 2 (i), which all but disappears for the larger values of n shown. ...
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We present an estimate of the power density function (spectrum) of binary K-th order spectral null codes. We work out the auto-correlation model in detail for second-and third-order spectral null codes. We compare the auto-correlation functions and spectra predicted by the model with those generated by full-set K-th order spectral null block codes.
... A binary code C is a qth-order spectral-null code withk information bits and length n (briefly, a q-OSN(n,k) code) if, and only if 1) C is a subset of SN (n, q), and 2) C has 2k codewords. When q = 1, these codes coincide with the so-called balanced or DC-free block codes [1]- [3], [5], [7], [10]- [14], [17]- [19], [21], [23]- [25], [28]. For values of q greater than 1, the q-OSN(n,k) codes are considered in digital recording to achieve a better rejection of the low frequency components of the transmitted signal and enhancing the error correction capability of codes used in partial-response channels [6], [14]. ...
... With regard to the complexity of Algorithm 4, note that step S0 can be accomplished in space O(k) memory bits and time O(k log k) bit operations by using any of the methods given in [1], [2], [7], [11], [12], [16], [18], [19], [21], and [29]. S1 can be accomplished in space O(r 5 ...
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A new efficient design of second-order spectral-null (2-OSN) codes is presented. The new codes are obtained by applying the technique used to design parallel decoding balanced (i.e., 1-OSN) codes to the random walk method introduced by some of the authors for designing 2-OSN codes. This gives new non-recursive efficient code designs, which are less redundant than the code designs found in the literature. In particular, if $k \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}}$ is the length of a 1-OSN code then the new 2-OSN coding scheme has length $n=k+r \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}}$ with an extra redundancy of $r\simeq 2\log _{2}k+(1/2)\log _{2}\log _{2} k-0.174$ check bits, with $k$ and $r$ even and $n$ multiple of 4. The whole coding process requires $O(k\log k)$ bit operations and $O(k)$ bit memory elements.
... The design of a code with power spectral density (PSD) zero at its DC-component, called DC-free codes [70], [71], becomes a necessity for AC coupling of the signal to the medium. The DC-balanced codes have found widespread applications in digital transmission and recording systems [72]- [73]. ...
... Besides the fact that DC-balanced codes offer a larger rejection of low-frequency components, the second derivative of the spectrum of a certain subclass is null at zero frequency. Such codes are called dc 2 -balanced codes [74], [71]. This class of codes has furthermore also been generalized for higher orders, and called Higher Order Spectral Null (HOSN) codes. ...
Article
In this paper, we present two public-key cry ptosystems over finite fields. First of them is based on polynomials. The presented system also considers a digital signature algorithm. Its security is based on the difficulty of finding discrete logarithms over GF(qd+1) with sufficiently large q and d. Is is also examined along with comparison with other polynomial based public-key systems. The other public-key cryptosystem is based on linear codes. McEliece studied the first code-based public-key cryptosystem. We are inspired by McEliece system in the construction of the new system. We examine its security using linear algebra and compare it with the other code-based cryptosystems. Our new cryptosystems are too reliable in terms of security.
Chapter
Baseband modulation codes are widely applied in such diverse fields as digital line transmission [21, 28, 18, 116], digital optical transmission [111, 16], and digital magnetic and optical storage [ 105, 100, 106]. They act to translate the source data sequence d n into a sequence a k that is transmitted across the channel (Fig. 4.1; see also Chapter 3). The principal goal is to enable the receiver to produce reliable decisions $${\hat d_n}$$ about d n . Code design should, therefore, account for the characteristics of both channel and receiver. (Because modulation coding is the only type of coding that we consider in this chapter, we shall usually say ‘coding’ where we mean ‘modulation coding’.)
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Tallini and Bose introduced a recursive method to construct second-order spectral-null codes which result in short ( information length 16) base optimal second-order spectral-null codes that can be easily implemented either with table lookup or enumerative coding. In this brief contribution, the extended base suboptimal second-order spectral-null codes ( 17 information length 32) obtained from two short base codes by lookup table are used to reduce the overall code length.
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