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$\ell $. 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.

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.

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, Kn, for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations ofKp+1and K2p, where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1 -factorization of Knis related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the Kp+1family of perfect 1 -factorization from the K2pfamily. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related.

Certain storage media such as flash memories use write-asymmetric, multi-level storage elements. In such media, data is stored in a multi-level memory cell the contents of which can only be increased, or reset. The reset operation is expensive and should be delayed as much as possible. Mathematically, we consider the problem of writing a binary sequence into write-asymmetric q-ary cells, while recording the last r bits written. We want to maximize t, the number of possible writes, before a reset is needed. We introduce the term Buffer Code, to describe the solution to this problem. A buffer code is a code that remembers the r most recent values of a variable. We present the construction of a single-cell (n=1) buffer code that can store a binary (l=2) variable with t=[q/2r-1]+r-2 and a universal upper bound to the number of rewrites that a single-cell buffer code can have: t ≤ [q-1/lr-1]·r+[logl{[(q-1) mod (lr- 1)]+1}]. We also show a binary buffer code with arbitrary n, q, r, namely, the code uses n q-ary cells to remember the r most recent values of one binary variable. The code can rewrite the variable t = (q-1)(n-2r+1)+r-1 times, which is asymptotically optimal in q and n. We then extend the code construction for the case r=2, and obtain a code that can rewrite the variable t=(q-1)(n-2)+1 times. When q=2, the code is strictly optimal.

Several physical effects that limit the reliability and performance of Multilevel Flash memories induce errors that have low magnitude and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions for such channels when the number of errors is bounded by t. The construction uses 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. An extension of the construction is proposed to include systematic codes as a benefit to practical implementation.

Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories and other non-volatile memories based on floating-gate cells have become a very important family of such memories. We model them by the Write Asymmetric Memory (WAM), a memory where each cell is in one of q states - state 0,1,..., q-1 - and can only transit from a lower state to a higher state. Data stored in a WAM can be rewritten by shifting the cells to higher states. Since the state transition is irreversible, the number of times of rewriting is limited. When multiple variables are stored in a WAM, we study codes, which we call floating codes, that maximize the total number of times the variables can be written and rewritten. In this paper, we present several families of floating codes that either are optimal, or approach optimality as the codes get longer. We also present bounds to the performance of general floating codes. The results show that floating codes can integrate the rewriting capabilities of different variables to a surprisingly high degree.

The existence of a perfect 1-factorization of the complete graph Kn, for arbitrary n, is a 40-year old open problem in graph theory. Two infinite families of perfect 1-factorizations are known for K2pand Kp+1, where p is a prime. It was shown in L. Xu et al. (1999) that finding a perfect 1-factorization of Kncan be reduced to a problem in coding, i.e. to constructing an MDS, lowest density array code of length n. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the Kp+1family of perfect 1-factorizations from the K2pfamily, by applying the reduction mentioned above. Namely, techniques from coding theory are used to prove a new result in graph theory

A linear threshold element computes a function that is a sign of a weighted sum of the input variables. The best known lower bounds on the size of threshold circuits are for depth-2 circuits with small (polynomial-size) weights. However, in general, the weights are arbitrary integers and can be of exponential size in the number of input variables. Namely, obtaining progress in lower bounds for threshold circuits seems to be related to understanding the role of large weights. In the present literature, a distinction is made between the two extreme cases of linear threshold functions with polynomial-size weights, as opposed to those with exponential-size weights. Our main contributions are in devising two novel methods for constructing threshold functions with minimal weights and filling up the gap between polynomial and exponential weight growth by further refining the separation. Namely, we prove that the class of linear threshold functions with polynomial-size weights can be divided into subclasses according to the degree of the polynomial. In fact, we prove a more general result---that there exists a minimal weight linear threshold function for any arbitrary number of inputs and any weight size.

The RAIN project is a research collaboration between Caltech and NASA-JPL on distributed computing and data-storage systems for future spaceborne missions. The goal of the project is to identify and develop key building blocks for reliable distributed systems built with inexpensive off-the-shelf components. The RAIN platform consists of a heterogeneous cluster of computing and/or storage nodes connected via multiple interfaces to networks configured in fault-tolerant topologies. The RAIN software components run in conjunction with operating system services and standard network protocols. Through software-implemented fault tolerance, the system tolerates multiple node, link, and switch failures, with no single point of failure. The RAIN-technology has been transferred to Rainfinity, a start-up company focusing on creating clustered solutions for improving the performance and availability of Internet data centers. In this paper, we describe the following contributions: 1) fault-tolerant interconnect topologies and communication protocols providing consistent error reporting of link failures, 2) fault management techniques based on group membership, and 3) data storage schemes based on computationally efficient error-control codes. We present several proof-of-concept applications: a highly-available video server, a highly-available Web server, and a distributed checkpointing system. Also, we describe a commercial product, Rainwall, built with the RAIN technology.

We present a class of array code of size n/spl times/l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture.