Single Byte Error Correcting—Double Byte Error Detecting Codes for Memory Systems

Musashino Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation
IEEE Transactions on Computers (Impact Factor: 1.38). 08/1982; DOI: 10.1109/TC.1982.1676056
Source: IEEE Xplore

ABSTRACT In a memory that uses byte-organized memory chips, each containing b (≥2) output bits, a single chip failure is likely to affect many bits within a byte. Single byte error correcting–double byte error detecting codes (SbEC–DbED codes) are used in this kind of memory system to increase reliability.

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    ABSTRACT: Burst errors are very common in practice. There have been many designs in order to control and correct such errors. Recently, a new class of byte error control codes called spotty byte error control codes has been specifically designed to fit the large capacity memory systems that use high-density random access memory (RAM) chips with input/output data of 8, 16, and 32 bits. The MacWilliams identity describes how the weight enumerator of a linear code and the weight enumerator of its dual code are related. Also, Lee metric which has attracted many researchers due to its applications. In this paper, we combine these two interesting topics and introduce the m-spotty generalized Lee weights and the m-spotty generalized Lee weight enumerators of a code over Z q and prove a MacWilliams type identity. This generalization includes both the case of the identity given in the paper [I. Siap, MacWilliams identity for m-spotty Lee weight enumerators, Appl. Math. Lett. 23 (1) (2010) 13-16] and the identity given in the paper [M. \"Ozen, V. \c{S}iap, The MacWilliams identity for m-spotty weight enumerators of linear codes over finite fields, Comput. Math. Appl. 61 (4) (2011) 1000-1004] over Z2 and Z3 as special cases.
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    ABSTRACT: Single byte (= b bits) error-correcting and double byte error-detecting (SbEC-DbED) codes are useful for semiconductor memory systems with memory chips of b-bit output per one address. In this paper, codes with information length k up to k = 128 bit for 2 b 8 are derived experimentally using a computer. As a result, codes with longer code lengths than before are obtained. In particular, as for the codes of b = 2 and k = 64, 12 check bits were required in the conventional codes, but the codes requiring only 10 check bits are derived by experiments and are an excellent result from a practical viewpoint too. In the derivation, each column in an H matrix of the code is regarded as an element in a Galois field GF (qr) where q = 2b (b is the number of bits in a byte) and r is the number of check bytes. to obtain codes with large code length, as many elements as possible are selected in GF(qr) satisfying the code condition. Here, a subset in a GF(qr) whose elements are linearly independent, is used to select the columns of the H matrix. Moreover, all the elements of GF(qr) arising as operation results in the derivation process can be in correspondence to the elements of the subset. Consequently, the objective sets are restricted and the derivation through computer experiments is facilitated.
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    ABSTRACT: It is a well established fact that m-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high-density RAM chips with wide I/O data (e.g. 8, 16 or 32 bits). Recently, a MacWilliams identity that establishes an important relation between an m-spotty weight enumerator of a binary code and its dual has been proven in Kazuyoshi Suzuki et al. (2007) [5]. In this paper, we introduce the m-spotty Lee weights and the m-spotty Lee weight enumerator of a quaternary code and prove a MacWilliams type identity.
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