We consider codes for channels with extreme noise that emerge in various low-power applications. Simple LDPC codes with parity checks of weight 3 are first studied for any code dimension m → ∞. These codes form modulation schemes: they improve the original channel outputs for any SNR > -6 dB (per information bit) and gain 3 dB over uncoded modulation as SNR grows. However, they also have a floor on the output bit error rate (BER) irrespective of their length. Tight lower and upper bounds, which are virtually identical to simulation results, are then obtained for BER at any SNR. We also study a combined scheme that splits $m$ information bits into $b$ blocks and protects each with some polar code. Decoding moves back and forth between polar and LDPC codes, every time using a polar code of a higher rate. For m → ∞and a sufficiently large parameter b, this design yields a vanishing BER at any SNR above the Shannon limit of -1.59 dB and has complexity order of m log m per information bit.

We combine polar and LDPC codes to address data correction for various low-power applications. We first use long low-rate LDPC codes that have parity checks of a low weight: Decoding performs several iterations of the belief propagation (BP) algorithm that recalculates the information bits only. Partially corrected bits are then passed to a short polar code that uses successive cancellation list (SCL) decoder. The newly corrected bits then serve as the new inputs for an LDPC decoder. For codes of rate less than 0.1, the algorithm performs on par with a CA-SCL decoder, while substantially reducing its latency.

We consider code design for high-noise memoryless channels, which emerge in various low-power applications, such as IoT or sensor networks. To address this case, we design simple LDPC-type codes that have growing dimension m and length m(m + 1)/ 2. These codes can be regarded as a "weakly-coded" modulation: they outperform uncoded modulation for the signal-to-noise ratios (SNR) above -3 dB per information bit and achieve a 3 dB gain as SNR grows. Similar to uncoded modulation, these codes also exhibit a floor on the output bit error rate (BER) for any m. To improve code performance, information bits are further protected with some polar code of length m. The overall design has low complexity of order m(2) log m and a vanishing BER of order exp{-m(1/2)}. It substantially outperforms biorthogonal codes for any SNR > 0 dB given the same code rate or blocklength.

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) > 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)

We study analytically and numerically decoding properties of finite-rate hypergraph-product quantum low density parity-check codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent X and Z errors. Several nontrivial lower and upper bounds for the decodable region are constructed analytically by analyzing the properties of the homological difference, equal minus the logarithm of the maximum-likelihood decoding probability for a given syndrome. Numerical results include an upper bound for the decodable region from specific heat calculations in associated Ising models and a minimum-weight decoding threshold of approximately 7%.

The techniques of distance verification known for general linear codes are re-applied to quantum stabilizer codes. Then distance verification is addressed for classical and quantum LDPC codes. New complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of LDPC codes. These bounds are expressed in terms of the erasure-correcting capacity of the corresponding ensemble. We also present a new irreducible-cluster technique that can be applied to any LDPC code and takes advantage of parity-checks' sparsity for both classical and quantum LDPC codes. This technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances, which also include all known families of quantum LDPC codes.

We survey the known list decoding algorithms for polar codes and compare their complexity. Index terms: Polar codes; Reed-Muller codes; successive cancellation decoding.

We consider recursive decoding techniques for RM codes, their subcodes, and newly designed codes. For moderate lengths up to 512, we obtain near-optimum decoding with feasible complexity.

We analyze successive cancellation (SC) decoder by using two random functions. The first function is related to the likelihoods of 0 and 1 in each code position, while the second gives the difference between their posterior probabilities. We then study the second power moments of both functions. We show that these moments are being squared in channel transformations, while their product tends to 0 for growing lengths $n$. This gives an elementary proof of polarization properties of SC decoding. We also derive a simple ordering of decoding channels with construction complexity of order $n\log n$.

We analyze successive cancellation (SC) decoder by using two random functions. The first function is related to the likelihoods of 0 and 1 in each code position, while the second gives the difference between their posterior probabilities. We then study the second power moments of both functions. We show that these moments are being squared in channel transformations, while their product tends to 0 for growing lengths $n$. This gives an elementary proof of polarization properties of SC decoding. We also derive a simple ordering of decoding channels with construction complexity of order $n\log n$.

We design polar codes of blocklength n→∞ and code rate R →1 that achieve the vanishing output error rates on the binary symmetric channels with transition error probability p → 0. These codes have a substantially smaller redundancy order (1 - R)n than do other known high-rate codes, such as Reed-Muller (RM) or BCH codes. The construction is explicit and has complexity of order nlog n. We also design asymptotically optimal low-rate codes that achieve the vanishing output error rates if p → 1/2.

Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube F-2(m). In this paper, we study punctured Reed-Muller codes Pr(m, b), whose positions are restricted to the m-tuples of a given Hamming weight b. In combinatorial terms, this paper concerns m-variate Boolean polynomials of any degree r, which are evaluated on a Hamming sphere of some radius b in F-2(m). Codes Pr (m, b) inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical b-layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes Pr (m, b) and find their distances for the arbitrary values of parameters r, m, and b. Finally, we describe recursive (successive cancellation) decoding of these codes.

The problem of finding code distance has been long studied for the generic ensembles of linear codes and led to several algorithms that substantially reduce exponential complexity of this task. However, no asymptotic complexity bounds are known for distance verification in other ensembles of linear codes. Our goal is to re-design the existing generic algorithms of distance verification and derive their complexity for LDPC codes. We obtain new complexity bounds with provable performance expressed in terms of the erasure-correcting thresholds of long LDPC codes. These bounds exponentially reduce complexity estimates known for linear codes.

We consider Plotkin-type constructions that perform a multi-step recursive decomposition of a given code into the shorter codes and are similar to polar and Reed-Muller RM codes. However, we end this decomposition process at the various short codes instead of the single information bits used as end nodes in polar design. We also employ maximum likelihood ML decoding of the end codes. Such a design can reduce the output error rates of polarised constructions on the moderate blocklengths. We also analyse the complexity-performance trade-offs in order to optimise code design.

We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum low-density parity-check codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.

We survey the existing techniques for calculating code distances of classical codes and apply these techniques to generic quantum codes. For classical and quantum LDPC codes, we also present a new linked-cluster technique. It reduces complexity exponent of all existing deterministic techniques designed for codes with small relative distances (which include all known families of quantum LDPC codes), and also surpasses the probabilistic technique for sufficiently high code rates.

We report a study on the optimization of ultra-high-resolution electron-beam lithography for nanoscale patterning with two separate lift-off processes using positive and negative resists; the optimized method is suitable for the emerging area of nano-magnetoelectronics. If used together, these high-aspect-ratio processes can achieve information cells with a diameter of 9 nm, a square pitch of 26 nm, and an etch depth of at least 50 nm, as required for recording densities greater than 3 Tbit/in(2), provided that 3D integration includes between 2 and 8 independent magnetic bits. Such effective patterning can be used to further develop magnetic bits packed for ultra-high-density disk recording and the emerging field of magnetic tunneling junctions for logic and memory applications.