Codes for High-Speed Memories III: Bit / Byte Error Control Codes in .NET

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Codes for High-Speed Memories III: Bit / Byte Error Control Codes
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Byte-organized semiconductor memory chips are widely used in today s digital systems. The usual errors occurred in these systems are soft errors induced by external noise, a particles, etc., which are apt to be manifested as random bit errors in byte-organized systems. Memory cell failure also results in random bit errors. Therefore designers of error control codes for byte-organized memory systems must take into account two types of errors, byte errors and bit errors. Strictly speaking, bit errors are a class of byte errors that corrupt exactly one bit within a byte, but such errors are usually called bit errors, and all others called byte errors. Based on the background above, this chapter deals with a class of practical codes that controls such errors as mixing byte errors with bit errors. These are abbreviated and designated as follows: 1. SEC-DED-SbED (or BED) codes. Single b-bit byte (or burst) error detecting SECDED codes. 2. SbEC-DED codes. Single b-bit byte error correcting and double-bit error detecting codes. 3. SbEC-DEC codes. Single b-bit byte error correcting and double-bit error correcting codes. 4. SbEC-(Sb S)ED codes. Single b-bit byte error correcting and single-byte plus single-bit error detecting codes. Here, A plus B error means A error and B error occurred simultaneously. Hence the above code 4 can detect both single-byte errors and single-bit errors occurred simultaneously. The SEC-DED-SbED codes have found many applications in computer memory systems, all with byte size b 4 bits. Some of these codes have been applied to recent microprocessor chips as well as to recent server systems.
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Code Design for Dependable Systems: Theory and Practical Applications, by Eiji Fujiwara Copyright # 2006 John Wiley & Sons, Inc.
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We recall here the de nitions of byte errors and burst errors. A word is divided into bytes of length b; a single-byte error is meant to be any number of errors con ned to one byte. On the other hand, a burst error of length b is any number of errors con ned to b adjacent positions. A burst of length b may begin on any bit position of the word and can spread over portions of two adjacent bytes. Also a code capable of detecting (or correcting) bursts of length b implies that it is capable of detecting (or correcting) all bursts with smaller than or equal to length b. For the purpose of this chapter, a byte always refers to a byte of length b, and a burst refers to a burst of length b or less. Since we do not cover multiple burst error control codes, the term burst here will mean a single burst only. The SEC-DED codes are not capable of correcting or detecting byte errors or burst errors. On the other hand, SbEC codes and SbEC-DbED codes can correct byte errors but require higher redundancy. From this consideration, codes are required to detect single-byte errors as well as to correct single-bit errors and to detect double-bit errors. The SEC-DED-SbED class of codes is practical from the standpoint of requiring small additional redundancy to the existing SEC-DED codes, and hence has become very popular in recent commercial applications [CHEN84, TSUC86, SUNM95].
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Burst Error Detecting SEC-DED Codes (SEC-DED-BED Codes)
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Some types of codes that meet the requirements mentioned above have been proposed in [BOSS78, REDD78, FUJI80a, VARA83]. It turns out, however, that all these proposed codes are equivalent to the same type of codes, namely single-burst error detecting SECDED code [KANE83]. Generally, the single-burst error detecting SEC-DED codes include single-byte error detecting SEC-DED codes as a special case. Therefore, more ef cient codes of single-byte error detecting SEC-DED codes, namely SEC-DED-SbED codes, will be studied in Subsection 6.1.3. Here we consider the single-byte error detecting SEC-DED codes. Then it will be shown that these codes are the single-burst error detecting SEC-DED codes, called SEC-DEDBED codes. This type of codes is discussed in Subsection 6.1.2. The reader should be careful to note and distinguish the abbreviations BED and SbED. The BED stands for singleburst (of length b) error detection, and SbED stands for single-byte (of length b) error detection. We begin with the case of two sets of errors, E1 and E2 , where E1 is the error set consisting of all single-bit errors and E2 is the error set consisting of all byte errors, T excluding single-bit errors. Hence E1 E2 ;. (; is the empty set.) Theorem 6.1 A linear code, described by the matrix H, corrects all errors in E1 and detects all errors in E2 , if and only if: S 1. E HT 6 0 for all E 2 fE1 E2 g; 2. Ei HT 6 Ej HT for all Ei ; Ej 2 E1 , 3. Ei HT 6 Ej HT for all Ei 2 E1 , and for all Ej 2 E2 . This theorem can be easily proved so that the conditions 1 and 2 are those for satisfying single-bit error correction, and the conditions 1 and 3 are for single-byte error detection. The codes satisfying all these conditions will be referred to as SEC-SbED codes.
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