1 Coding Schemes for Multi-Level Channels with Unknown Gain and/or Offset Kees A. Schouhamer Immink Turing Machines Inc, Willemskade 15b-d, 3016 DK Rotterdam, The Netherlands E-mail: immink@turing-machines.com Summary - 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. Key words: Constant composition code, permutation code, flash memory, optical recording I. I NTRODUCTION We consider a communication codebook, S, of chosen q-ary sequences x =(x 1 ,x 2 ,...,x n ) over the q-ary alphabet Q = {0, 1,...,q - 1}, q ≥ 2, where n, the length of x, is a positive integer. Usually it is assumed that the primary distortion of a sent codeword x is additive noise. Here, however, it is assumed that the received vector, r, where r = a(x + ν )+ b, a> 0, is scaled by an unknown positive scaling factor (gain), a, and offsetted by an unknown offset, b, (that is, both quantities are unknown to both sender and receiver), where a and b are real numbers, and corrupted by additive noise ν = (ν 1 ,...,ν n ). Examples of such channels in practice are optical disc, where the gain and offset depend on the reflective index of the disc surface and the dimensions of the written features [1]. Reading errors in solid-state (Flash) memories may originate from low memory endurance, by which a drift of threshold levels in aging devices may cause programming and read errors [2], [3]. Mismatch between receiver and channel may lead to serious degradation of the error performance. There have been a few proposals to improve the detection process by making it less dependent on the channel’s actual gain and offset. We may introduce automatic gain (AGC) and offset control, which depends on the weighted average gain and offset of previously received codewords. In most channels, gain and offset are time-variant so that the gain and offset control may be suboptimal. We may use parts of the memory cell array as reference cells. The reference cells are written with known signal levels, and are continuously monitored to obtain estimates of the channel’s momentary gain and offset. The estimated offset values will be used to calibrate the threshold levels. In addition, codes have been used in practice to make the detection quality less dependent of the channel’s gain and offset. In optical disc recording devices, dc-balanced binary codes have been used to counter the effects of unknown gain and offset levels [4]. Jiang et al. [2] addressed a coding tech- nique called rank modulation for circumventing the difficulties in flash memories having aging offset levels. In this paper, we will study maximum likelihood detection of q-ary codewords stored in a solid-state memory whose gain and/or offset may change over time, and are assumed to be unknown to the receiver. We will investigate the properties of codewords that will make them immune against uncertainty in the channel’s gain and/or offset. Two quantities, as we will show later, play a key role in the design of such intrinsic resistance codes, namely the balance d c (x)= n  i=1 x i , and the energy d p (x)= n  i=1 x 2 i , which depend on the codeword x. We will consider codes, S, where d c (x) and d p (x) are constant with respect to x in S. In Section II, we will show that codes consisting of code- words that satisfy a balance (B), or energy (E), or a combina- tion of both (BE) constraints, are intrinsically resistant against the channel’s unknown gain and/or offset. In Section III, we will study properties of the set S of constrained codewords. We will, in Sections IV and V, compute the size of S using generating functions, and find, using statistical arguments, an approximate expression of the redundancy of BE constrained channels for asymptotically large values of n. In Section VI, we will describe our conclusions. II. I NTRINSICALLY RESISTANT CODES We select a codebook, S, of codewords, x, that all satisfy the same (E)nergy constraint n  i=1 x 2 i = d p , 978-1-4799-0446-4/13/$31.00 ©2013 IEEE 2013 IEEE International Symposium on Information Theory 709