Almost-Certainly Runlength-Limiting Codes

Standard runlength-limiting codes – nonlinear codes defined by trellises – have the disadvantage that they disconnect the outer errorcorrecting code from the bit-by-bit likelihoods that come out of the channel. I present two methods for creating transmissions that, with probability extremely close to 1, both are runlength-limited and are codewords of an outer linear error-correcting code (or are within a very small Hamming distance of a codeword). The cost of these runlength-limiting methods, in terms of loss of rate, is significantly smaller than that of standard runlength-limiting codes. The methods can be used with any linear outer code; low-density parity-check codes are discussed as an example. The cost of the method, in terms of additional redundancy, is very small: a reduction in rate of less than 1% is sufficient for a code with blocklength 4376 bits and maximum runlength 14. This paper concerns noisy binary channels that are also constrained channels, having maximum runlength limits: the maximum number of consecutive 1s and/or 0s is constrained to be r. The methods discussed can also be applied to channels for which certain other long sequences are forbidden, but they are not applicable to channels with minimum runlength constraints such as maximum transition-run constraints. I have in mind maximum runlengths such as r = 7, 15, or 21. Such constraints have a very small effect on the capacity of the channel. (The capacity of a noiseless binary channel with maximum runlength r is about 1 − 2.) There are two simple ways to enforce runlength constraints. The first is to use a nonlinear code to map, say, 15 data bits to 16 transmitted bits [8]. The second is to use a linear code that is guaranteed to enforce the runlength constraints. The disadvantage of the first method is that it separates the outer error-correcting code from the channel: soft likelihood information may be available at the channel output, but once this information has passed through the inner decoder, its utility is degraded. The loss of bit-by-bit likelihood information can decrease the performance of a code by about 2 dB [3]. The second method may be feasible, especially if low-density parity-check codes are used, since they are built out of simple parity constraints, but it only gives a runlength limit r smaller than 16 if the outer code’s rate is smaller than the rates that are conventionally required for magnetic recording (0.9 or so) [7]. I now present two simple ideas for getting the best of both worlds. The methods presented involve only a small loss in communication rate, and they are compatible with the use of linear error-correcting codes. The methods do not give an absolute guarantee of reliable runlength-limited communication; rather, as in a proof of Shannon’s noisy channel coding theorem, we will be able to make a statement like ‘with probability 1−10, this method will communicate reliably and satisfy the r = 15 runlength constraint’. This philosophy marries nicely with modern developments in magnetic recording, such as (a) the use of low-density parity-check codes, which come without cast-iron guarantees but work very well empirically [7]; and (b) the idea of digital fountain codes [1], which can store a large file on disc by writing thousands of packets on the disc, each packet being a random function of the original file, and the original file being recoverable from (almost) any sufficiently large subset of the packets – in which case occasional packet loss is unimportant. The ideas presented here are similar to, but different from, those presented by Immink [4–6], Deng and Herro [2], and Markarian et al. [9].