Conference Paper

Coding schemes for multi-level channels with unknown gain and/or offset

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Abstract

We will present coding techniques for transmission and storage channels with unknown gain and/or offset. It will be shown that a codebook of length-n q-ary codewords, S, where all codewords in S have equal balance and energy show an intrinsic resistance against unknown gain and/or offset. Generating functions for evaluating the size of S will be presented. We will present an approximate expression for the code redundancy for asymptotically large values of n.

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... (ii) Up to now, various coding techniques have been applied to alleviate the detection in case of channel mismatch, such as, rank modulation [32], balanced codes [33][34][35][36][37], and composition check codes [38]. ...
... The notion of dynamic thresholds based on balanced codes is introduced in [33] for the reading of binary sequences. It is further shown to be highly effective against errors caused by voltage drift in Flash memories [34][35][36]. A balanced code consists of the sequences where the number of ones equals the number of zeros. ...
... Gain and offset mismatch have a significant bearing on the error performance of MED asx related terms are dependent on a and b. In the prior art, constrained codes, specifically, d c/d c 2 −bal anced codes, are considered to counter the effects of gain and offset mismatch [36]. By definition, all codewords x in a dc/dc 2balanced code satisfy that the symbol sum n i =1 x i = a 1 and symbol energy n i =1 x 2 i = a 2 , are prescribed, where a 1 and a 2 are two positive integers selected by the code designer. ...
... Clearly, the redundancy of the method is two symbols per codeword. In a second prior art method, codes satisfying equal balance and energy constraints [8], which are immune to gain and offset mismatch, have been advocated. The redundancy of these codes, denoted by r 0 , is given by [8] r 0 ≈ log q n + log q (q 2 − 1) q 2 − 4 + log q π 12 √ 15 . ...
... In a second prior art method, codes satisfying equal balance and energy constraints [8], which are immune to gain and offset mismatch, have been advocated. The redundancy of these codes, denoted by r 0 , is given by [8] r 0 ≈ log q n + log q (q 2 − 1) q 2 − 4 + log q π 12 √ 15 . ...
... where φ( j ) is Euler's totient function that counts the totatives of j , i.e., the positive integers less than or equal to j that are relatively prime to j . We have computed the cardinalities of N 1 (q, n), N 2 (q, n), and P q,n by invoking (7), (8), and the expressions in Table I. Table II lists the results of our computations for selected values of q and n. ...
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... For evaluating (7), the decoder requires |S| = 2 n − 1 computations of δ(r,x) plus comparisons, which makes the new method unattractive for very large n. It is shown in [6] that the (time) complexity of the prior art method based on (3) can be reduced to n computations and comparisons using Slepian's method [14]. ...
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This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating functions, labelled enumeration and exponential generating functions, and finally multivariate enumeration and generating functions. It is largely oriented towards applications of combinatorial enumeration to random discrete structures and discrete mathematics models, as they appear in various branches of science, like statistical physics, computational biology, probability theory, and, last not least, computer science and the analysis of algorithms.
Analytic Combinatorics
  • P Flajonet
  • R Sedgewick
P. Flajonet and R. Sedgewick, 'Analytic Combinatorics', ISBN 978-0521-89806-5, Cambridge University Press, 2009.