##### From Code Design for Dependable Systems

9.4   SINGLE-BIT ERROR CORRECTING AND SINGLE-BYTE ERROR
LOCATING (SEC-Se=bEL) CODES

We study here another class of error locating codes, namely Single-bit Error Correcting
and Single e-bit (within a b-bit byte) Error Locating codes, or SEC-Se/bEL codes.

9.4.1   Code Conditions and Bounds

Let Es be the error set consisting of all single-bit errors, and let Ei(Ej) be the set of e
or fewer bits errors in the i( j)-th byte excluding single-bit errors, where e < b and b is
the byte length. Thus Ei Ç Es = Ej Ç Es = φ for all ij, where f is the empty set. The
following theorem describes the necessary and sufficient conditions that characterize
SEC-Se/bEL codes.

Theorem 9.10   A linear code, described by the parity-check matrix H, corrects all
errors in Es and locates all errors in Ei, or Ej , where i ≠ j, if and only if:

where HT means the transpose of H.

Proof   It is apparent that conditions 1 and...

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Memory Modules
Memory modules are computer chips used to add memory to a computer.
Parity Checkers and Generators
Parity checkers are integrated circuits (ICs) used in digital systems to detect errors when streams of bits are sent from a transmitter to a receiver. Parity generators calculate the parity of data packets and add a parity amount to them.
Error Correction Chips
Error correction chips are semiconductor ICs that are used to check and to correct data transmission errors.
SCSI adapters and SCSI controllers (SCSI cards) are computer interface cards that are installed in an expansion slot. They are used to connect the SCSI system to several devices and peripherals using a daisy chain method.
RAID Disk Arrays
RAID disk arrays are collections of storage disks with integrated controllers to manage the storage of data.

##### Topics of Interest

9.4.2 Design for SEC-Se/bEL Codes The tensor product of the error correcting codes and the error detecting codes is also applied in the design of SEC-Se/bEL codes. That is, tensor product of an SbEC...

9.3   SINGLE-BIT ERROR CORRECTING AND SINGLE-BLOCK ERROR LOCATING (SEC-Sb/p×bEL) CODES This section deals with the SEC-Sb/p×bEL codes, where B = p × b that correct single-bit...

9.1   ERROR LOCATION OF FAULTY PACKAGES AND FAULTY CHIPS As was shown in Figure 1.11 in Subsection 1.4.1, error location falls midway between the functions of error correction and error...

9.4.3   Evaluation Code Length   Figure 9.9 illustrates the relation between the information-bit length and the check-bit length of the SEC-Se/8EL codes for e = 2 and 6. In the figure the...

9.3.2   Design for SEC-Sb/p×bEL Codes 1. Codes Designed by Tensor Product — Codes I — In general, we can design the error locating codes by means of the tensor product of two...

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