Shaping codes are used to generate code sequences in which the symbols obey a prescribed probability distribution. They arise naturally in the context of source coding for noiseless channels with unequal symbol costs. Recently, shaping codes have been proposed to extend the lifetime of flash memory and reduce DNA synthesis time. In this paper, we study a general class of shaping codes for noiseless finite-state channels with cost and i.i.d. sources. We establish a relationship between the code rate and minimum average symbol cost. We then determine the rate that minimizes the average cost per source symbol (total cost). An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source.

In this paper, we investigate the use of recurrent neural network (RNN)-based detection of magnetic recording channels with inter-symbol interference (ISI). We refer to the proposed detection method, which is intended for recording channels with partial-response equalization, as Partial-Response Neural Network (PR-NN). We train bi-directional gated recurrent units (bi-GRUs) to recover the ISI channel inputs from noisy channel output sequences and evaluate the network performance when applied to continuous, streaming data. The computational complexity of PR-NN during the evaluation process is comparable to that of a Viterbi detector. The recording system on which the experiments were conducted uses a rate-2/3, (1,7) runlength-limited (RLL) code with an E2PR4 partial-response channel target. Experimental results with ideal PR signals show that the performance of PR-NN detection approaches that of Viterbi detection in additive white gaussian noise (AWGN). Moreover, the PR-NN detector outperforms Viterbi detection and achieves the performance of Noise-Predictive Maximum Likelihood (NPML) detection in additive colored noise (ACN) at different channel densities. A PR-NN detector trained with both AWGN and ACN maintains the performance observed under separate training. Similarly, when trained with ACN corresponding to two different channel densities, PR-NN maintains its performance at both densities. Experiments confirm that this robustness is consistent over a wide range of signal-to-noise ratios (SNRs). Finally, PR-NN displays robust performance when applied to a more realistic magnetic recording channel with MMSE-equalized Lorentzian signals.

For DNA data storage to become a feasible technology, all aspects of the encoding and decoding pipeline must be optimized. Writing the data into DNA, which is known as DNA synthesis, is currently the most costly part of existing storage systems. As a step toward more efficient synthesis, we study the design of codes that minimize the time and number of required materials needed to produce the DNA strands. We consider a popular synthesis process that builds many strands in parallel in a step-by-step fashion using a fixed supersequence S. The machine iterates through S one nucleotide at a time, and in each cycle, it adds the next nucleotide to a subset of the strands. The synthesis time is determined by the length of S. We show that by introducing redundancy to the synthesized strands, we can significantly decrease the number of synthesis cycles. We derive the maximum amount of information per synthesis cycle assuming S is an arbitrary periodic sequence. To prove our results, we exhibit new connections to cost-constrained codes.

In this paper, we investigate the use of recurrent neural network (RNN)-based detection of magnetic recording channels with inter-symbol interference (ISI). We refer to the proposed detection method, which is intended for recording channels with partial-response equalization, as Partial-Response Neural Network (PR-NN). We train bi-directional gated recurrent units (bi-GRUs) to recover the ISI channel inputs from noisy channel output sequences and evaluate the network performance when applied to continuous, streaming data. The recording system on which the experiments were conducted uses a rate-2/3, (1,7) runlength-limited (RLL) code with an E2PR4 partial-response channel target. Experimental results with ideal PR signals show that the performance of PR-NN detection approaches that of Viterbi detection in additive white gaussian noise (AWGN). Moreover, the PR-NN detector outperforms Viterbi detection and achieves the performance of Noise-Predictive Maximum Likelihood (NPML) detection in additive colored noise (ACN).

Shaping codes are used to encode information for use on channels with cost constraints. Applications include data transmission with a power constraint and, more recently, data storage on flash memories with a constraint on memory cell wear. In the latter application, system requirements often impose a rate constraint. In this paper, we study rate-constrained fixed-to-variable length shaping codes for noiseless, memoryless costly channels and general i.i.d. sources. The analysis relies on the theory of word-valued sources. We establish a relationship between the code expansion factor - the ratio of the expected codeword length to the length of the input source word - and the minimum average symbol cost. We then determine the expansion factor that minimizes the average cost per source symbol (total cost), corresponding to a conventional optimal source code with cost. An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source. Shaping codes often incorporate, either explicitly or implicitly, some form of non-equiprobable signaling. We use our results to further explore the connections between shaping codes and codes that map a sequence of i.i.d. source symbols into an output sequence of symbols that are approximately independent and distributed according to a specified target distribution, such as distribution matching (DM) codes. Optimal DM codes are characterized in terms of a new performance measure - generalized expansion factor (GEF) - motivated by the costly channel perspective. The GEF is used to study DM codes that minimize informational divergence and normalized informational divergence.

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances, and we discuss decoding algorithms and correctable patterns of errors and erasures. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes, generalizing a recent construction of capacity-achieving rate-compatible polar codes. Applications of rate-compatible codes to data storage are considered. We design two-level rate-compatible codes based on Bose-Chaudhuri-Hocquenghem (BCH) and low-density parity-check (LDPC) codes which are two popular codes widely used in the data storage industry, and then we evaluate the performance of these codes in multi-level cell (MLC) flash memories. We also examine code performance on binary and $q$ -ary symmetric channels. Finally, we briefly discuss two variations of our main construction and their relative performance.

Data shaping is a coding technique that has been proposed to increase the lifetime of flash memory devices. Several data shaping codes have been described in recent work, including endurance codes [2] and direct shaping codes for structured data [8], [4], [5]. In this paper, we study information-theoretic properties of a general class of data shaping codes and prove a separation theorem stating that optimal data shaping can be achieved by the concatenation of optimal lossless compression with optimal endurance coding. We also determine the expansion factor that minimizes the total wear cost. Finally, we analyze the performance of direct shaping codes and establish a condition for their optimality.

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes.

In this work, we study data shaping codes for flash memory. We first review a recently proposed direct shaping code for SLC (one bit per cell) flash memory that reduces wear by minimizing the average fraction of programmed cells. Then we describe an adaptation of this algorithm that provides data shaping for MLC (two bits per cell) flash memory. It makes use of a page- dependent cost model and is designed to be compatible with the standard procedure of row-by- row, page-based, wordline programming. We also give simulation results demonstrating the performance of these data shaping codes when applied to English language text. Finally, we use a random walk model to analyze the potential error propagation properties of direct shaping codes when used in a noisy memory.