We discuss the problem of finding a minimal aliasing code among a class of systematic single-error-correction codes that are suitable to be implemented within DRAM die, as opposed to external ECC used in memory controller outside of DRAM chip. We prove a sharp lower bound of aliasing probability and propose a simple method to come up with a code that meets the bound. By an experiment, we also demonstrate that a randomly chosen code is likely to have much more aliasings with overwhelmingly high probability.

Deep learning has achieved remarkable success across a wide range of applications, such as language modeling, computer vision, recommendation systems, and robotics. However, the growing size of models and their increasing computational demands pose significant challenges, particularly for resource-constrained devices. A promising approach to address these challenges is extreme quantization, exemplified by binary and ternary neural networks. These techniques significantly reduce model size by quantizing weights and activations to 1 bit or 1.58 bits, while simplifying computation, making them well-suited for efficient deployment in resource-limited environments. This paper presents a comprehensive review of extreme quantization techniques, organized into three key areas: (1) a comparative analysis of quantizing only the weights (e.g., binary weight networks, ternary weight networks) versus quantizing both weights and activations (e.g., binary neural networks, ternary neural networks), along with a discussion of the progress and trade-offs of their approaches; (2) an examination of how extreme quantization, initially applied to convolutional neural networks, has been extended to Transformer architectures; and (3) an overview of compute-in-memory architectures optimized for binarization and ternarization, including designs based on advanced bit-cell technologies.

An m-extracting procedure produces unbiased random bits from a loaded dice with m faces, and its output rate, the average number of output per input is bounded by the Shannon entropy of the source. This information-theoretic upper bound can be achieved only asymptotically as the input size increases, by certain extracting procedures that we call asymptotically optimal. Although a computationally efficient asymptotically optimal 2-extracting procedure has been known for a while, its counterparts for m-ary input, m > 2, was found only recently, and they are still relatively complicated to describe. A binarization takes inputs from an m-faced dice and produce bit sequences to be fed into a binary extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an m-extracting procedure via a binarization. A binarization is to be called complete, if it preserves the asymptotic optimality, and such a procedure has been proposed by Zhou and Bruck. We show that a complete binarization naturally arises from a binary tree with m leaves. Therefore, there exist complete binarizations in abundance and Zhou-Bruck scheme is an instance of them. We now have a relatively simple way to obtain an asymptotically optimal and computationally efficient m-extracting procedure, from a binary one, because these binarizations are both conceptually and computationally simple. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments.

This paper proposes two new encoders for data bus inversion (DBI), which conventionally uses a majority voter to pick a data representation that minimizes switching activities and thus reduces the corresponding energy consumption. The new encoders employ simpler approximate voters comprising only two gate levels, which improve latency more than twice while still achieving switching activity savings by 9% and 11%, respectively. Although the proposed voters are not always accurate, the errors in the voters do not affect the correctness of data movement. We report various metrics, including latencies, areas, and operating powers, regarding five different designs, two proposed designs along with three conventional designs, based on 65-nm process implementations.

Data bus inversion (DBI) is an encoding technique that saves power in data movement in which the majority function plays an essential role. For a latency optimization, the majority function can be replaced by a majority approximator that allows for a small error in majority voting to obtain a faster encoder that still saves power. In this work, we propose two systematic approaches for finding high-performance majority approximators. First, we perform an exhaustive search of all possible Boolean functions to find an optimal approximator based on a certain circuit structure comprised of fifteen logic gates. The approximator found by the systematic search can be implemented using compound gates, resulting in a latency-efficient design with only two gate levels. Compared with prior works using a heuristic idea, the proposed circuit runs at the same speed but achieves greater switching activity savings. Second, we propose another majority approximator using the average of three randomly permuted copies of the approximator found in the first approach. We show that the second proposed approximator achieves even higher savings in switching activity as its function is closer to a true majority voter. We report various performance metrics of the newly found majority approximators based on syntheses using a 65 nm process.

Peres algorithm applies the famous von Neumann trick recursively to produce unbiased random bits from biased coin tosses. Its recursive nature makes the algorithm simple and elegant, and yet its output rate approaches the information-theoretic upper bound. However, it is relatively hard to explain why it works, and it appears partly due to this difficulty that its generalization to many-valued source was discovered only recently. Binarization tree provides a new conceptual tool to understand the innerworkings of the original Peres algorithm and the recently-found generalizations in both aspects of the uniform random number generation and asymptotic optimality. Furthermore, it facilitates finding many new Peres-style recursive algorithms that have been arguably very hard to come by without this new tool.