Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories

Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation

Rewriting Codes for Joint Information Storage in Flash Memories

Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories have become a very important family of such memories. A flash memory cell has q states—state 0.1.....q-1 - and can only transit from a lower state to a higher state before the expensive erasure operation takes place. We study rewriting codes that enable the data stored in a group of cells to be rewritten by only shifting the cells to higher states. Since the considered state transitions are irreversible, the number of rewrites is bounded. Our objective is to maximize the number of times the data can be rewritten. We focus on the joint storage of data in flash memories, and study two rewriting codes for two different scenarios. The first code, called floating code, is for the joint storage of multiple variables, where every rewrite changes one variable. The second code, called buffer code, is for remembering the most recent data in a data stream. Many of the codes presented here are either optimal or asymptotically optimal. We also present bounds to the performance of general codes. The results show that rewriting codes can integrate a flash memory’s rewriting capabilities for different variables to a high degree

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.Comment: 26 page

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function (PUF) response. We extend classical coding methods to construct multiply constant-weight codes from known $q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and are shown to be asymptotically tight to a constant factor under certain conditions. We also examine the rates of the multiply constant-weight codes and interestingly, demonstrate that these rates are the same as those of constant-weight codes of suitable parameters. Asymptotic analysis of our code constructions is provided

A Computational Framework for Efficient Error Correcting Codes Using an Artificial Neural Network Paradigm.

The quest for an efficient computational approach to neural connectivity problems has undergone a significant evolution in the last few years. The current best systems are far from equaling human performance, especially when a program of instructions is executed sequentially as in a von Neuman computer. On the other hand, neural net models are potential candidates for parallel processing since they explore many competing hypotheses simultaneously using massively parallel nets composed of many computational elements connected by links with variable weights. Thus, the application of modeling of a neural network must be complemented by deep insight into how to embed algorithms for an error correcting paradigm in order to gain the advantage of parallel computation. In this dissertation, we construct a neural network for single error detection and correction in linear codes. Then we present an error-detecting paradigm in the framework of neural networks. We consider the problem of error detection of systematic unidirectional codes which is assumed to have double or triple errors. The generalization of network construction for the error-detecting codes is discussed with a heuristic algorithm. We also describe models of the code construction, detection and correction of t-EC/d-ED/AUED (t-Error Correcting/d-Error Detecting/All Unidirectional Error Detecting) codes which are more general codes in the error correcting paradigm