This paper presents a state-of-the-art review of error-correcting codes for computer semiconductor memory applications. The construction of four classes of error-correcting codes appropriate for semiconductor memory designs is described, and for each class of codes the number of check bits required for commonly used data lengths is provided. The implementation aspects of error correction and error detection are also discussed, and certain algorithms useful in extending the error-correcting capability for the correction of soft errors such as α-particle-induced errors are examined in some detail.

Error-correcting codes for semiconductor memory applications: a state-of-the-art review

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{this http URL}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.

Tables of subspace codes

Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed---Muller codes, and even further to LDPC codes, random network codes, and distributed storage. This article reviews briefly the known links, and then focuses on new links and new directions. We present new results and open problems to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.

Galois geometries and coding theory

IBM's 6250 bpi 3420 series tape units require a powerful error-correcting code for the standard 9-track format. The optimal rectangular code (0RC), presented here, is designed to correct any single-track error or, given erasure pointers, any double-track error in the tape. The code achieves this by conforming to a rectangular codeword of which two orthogonal sides are check bits. The code is specially tailored from a general class of b-adjacent codes. The ORC can be implemented without a buffer for encoding and offers a simple error-correction mechanism. The code can be generalized to multiple-channel applications. Introduction Models 4, 6 and 8 of the IBM 3420 series tape units record 6250 bits per inch (bpi), one of the highest densities commercially available on standard !-inch 9 track tapes. To achieve this density, a number of stringent engineering requirements had to be met. Included was a fast and powerful error-correcting scheme. The standard !-inch 9-track tape system evolved through the extensive use of tapes over many years. The most frequently used 8-bit byte of information and a parity check bit are accommodated vertically in these 9 tracks. The ninth parity track is usually called the VRC (vertical redundancy check) track. In low density recording such as IBM 729 series tapes, this VRC and a longitudinal parity byte (LRC) at the end of a record were sufficient. As the bit density increased, another check byte called CRC (cyclic redundancy check) was added at the end of the record to provide track error correction such as in IBM's 800 bpi tape units [1]. In designing new tape products, compatibility with the existing 9-track standard data format is one of the prime considerations if tapes are to be recorded in different machines and interchanged freely. The new products, moreover, require a superior error-correction code because of increased bit densities and I or tape speeds and other similar reasons. The new coding scheme presented in this paper is designed to satisfy these requirements. Bit density along the direction of tape motion is conventionally much higher than that across the tape. The ratio has steadily increased from about 40 in 800 bpi tapes to two to three orders of magnitude in the current high density tape machines. As a result, most common errors are track erasures. The erroneous tracks are often identified by the loss of signal in the read amplifiers, and lor excessive phase shift in clock and detection circuits or other similar electronic indicators. The coding scheme of this paper is designed to correct this type of track error or erasure. The same scheme, moreover, corrects many random errors which may be spread over many tracks in a record. The codewords are in the form of a rectangle with two orthogonal sides as check bits, and they resemble the diagonal check scheme [2]. The encoding and decoding, however, are carried out in an algebraic manner and the redundancy is minimal. The erroneous bits along a track form a cluster-error that can be corrected as a unit in each rectangular codeword. Each rectangular codeword is processed on the fly, independent of whether cluster errors are from the same track or different tracks. It is well known that the error-correcting codes for symbols from GF(2 ) , the galois field of 2 elements, can be used for correction of clusters of b-adjacent binary symbols. The generalized Hamming codes [3-5], Reed-Solomon codes [6], and BCH codes [4] with symbols from GF(2 ) are some of the examples. These codes can be described in a binary check matrix as proposed by Cocke [7]. Bossen [3] shows a high-speed implementation using the binary matrix description. Other methods of implementing the GF(2 ) codes for 579 NOVEMBER 1974 MAGNETIC TAPE CODINO

Optimal rectangular code for high density magnetic tapes

The error-correcting codes for symbols from GF (2b) are often used for correction of byte-errors in binary data. In these byte-error-correcting codes each check symbol in GF (2b) is expressed as b binary check digits and each information symbol in GF (2b), likewise, is expressed by b binary information digits. A new class of codes for single-byte-error correction is presented. The code is general in that the code structure does not depend on symbols from GF (2b). In particular, the number of check bits are not restricted to the multiples of b as in the case of the codes derived from GF (2b) codes. The new codes are either perfect or maximal and are easily implementable using shift registers.

A.M. Patel

Papers

A General Class of Maximal Codes ror Computer Applications