J.W. Schwartz and J.K. Wolf (ibid., vol. 39, no. 11, pp. 1403-1404, Nov. 1990) gave a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects eight of the nine double adjacent errors within any of the three 4-bit nibbles. We present a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects any pair of errors within a nibble

Blaum, Mario

Bruck, Jehoshua

Tolhuizen, Ludo

English

Caltech Authors - Main

125 IEEE TRANSACTIONS ON COMPUTERS, VOL. 43, NO. 1, JANUARY 1994 
Comments 
A Note on “A Systematic (12,8) Code for Correcting 
Single Errors and Detecting Adjacent Errors” 
Mario Blaum, Jehoshua Bruck, and Ludo Tolhuizen 
Abstract-Schwartz and Wolf give a parity check matrix for a system- 
atic (12,8) binary code that corrects all single errors and detects eight of 
the nine double adjacent errors within any of the three 4-bit nibbles. Here, 
we present a parity check matrix for a systematic (12,8) binary code that 
corrects all single errors and detects any pair of errors within a nibble. 
nibble, parity-check matrix, systematic codes. 
Zndex Terms-Adjacent errors, error-correcting/detecting codes, 4-bit 
In [3], the authors studied the following problem: Given a byte 
stored as three 4-bit nibbles, such that 8 of the bits carry information 
and 4 are redundant, encode the information bits in such a way that 
any single error will be corrected and any two adjacent bits in error 
in a nibble will be detected. The authors come up with a solution 
that in fact corrects any single error and detects 8 out of the 9 cases 
of double adjacent errors in a nibble. 
Note that the code given in [3] is not optimal-it can detect only 
8 out of the 9 patterns of double adjacent errors within a nibble. 
Moreover, the authors in [3] claim that it is not possible to find a 
systematic code that can detect the 9 patterns of double adjacent 
errors within a nibble. This claim, however, is true for the particular 
case in which the first 8 bits carry the information and the last 4 
bits carry the redundancy. Any code equivalent to this one (Le., its 
coordinates are a permutation of the coordinates of the original code) 
is also systematic [2]. 
In this note, we present a counterexample to the claim in [3], 
namely, a systematic (12,8) code that can correct any single error 
and detect any of the 9 patterns of double adjacent errors within 
a nibble. Moreover, our code detects any pattern of double errors 
(adjacent or not) within a nibble. 
As pointed out above, an ( t i .  k )  code is systematic if there are 
k positions in which the codewords carry the information, and the 
remaining ) t  - k positions are redundant [2]. Consider a (12,8) code 
with the following parity check matrix: 
1 1 0 0  1 0 0 1  l O l 0 )  
1 0 1 0  1 0 1 0  1 0 0 1  
1 0 0 1  1 1 0 0  0 0 1 1 ’  
1 0 0 1  0 0 1 1  1 1 0 0  
H =  ( 
Since the columns are all distinct, this code is single-error cor- 
recting [2]. As we can see, the 4 x 4 submatrix of H at locations 
2, 3, 6, and 10 is the identity matrix. Therefore, these coordinates 
correspond to the redundancy and the information is carried in bits 
1, 4, 5,  7, 8, 9, 11, and 12. 
Manuscript received May 7, 1991. 
M. Blaum and J .  Bruck are with IBM Research Division, Almaden Research 
L. Tolhuizen is with Philips Research Laboratories, 5600 JA Eindhoven, 
IEEE Log Number 9209032. 
Center, San Jose, CA 95120. 
The Netherlands. 
Notice that the syndromes corresponding to double errors within 
nibbles (Le., the EXCLUSIVE-OR’s of all possible pairs of columns 
of H within a nibble) are 0111, 1100, or 1011. These vectors do  
not correspond to any of the columns of H ,  therefore, they cannot be 
confused with a syndrome corresponding to a single error. Hence, the 
code can correct any single error and detect any double error within 
a nibble, providing a counterexample to [3]. Let us point out that our 
results have been generalized in a recent paper [l]. 
REFERENCES 
T. Etzion, “Optimal codes for correcting single errors and detecting adja- 
cent errors,’’ IEEE Trans Inform. Theory, vol. 38, no. 4, pp. 1357-1360, 
July 1992. 
J. H. van Lint, Introduction to Coding Theory. New York: Springer- 
Verlag, 1982. 
J. W. Schwartz and J .  K. Wolf, “A systematic (12,8) code for correcting 
single errors and detecting adjacent errors,” IEEE Trans. Comput., vol. 
39, no. 11, pp. 1403-1404, Nov. 1990. 
Comments on “Synthetic Traces for ’kace-Driven 
Simulation of Cache Memories”’ 
Syed Masud Mahmud 
Abs t rac t4  number of errors have been discovered in the above paper’ 
The authors of paper’ have corrected some of these errors and presented 
in correspondence? The remaining errors are corrected and presented in 
this correspondence. 
Zndex Terms- Cache memories, performance analysis, trace-driven 
simulation, LRU stack model, program locality. 
I. INTRODUCTION 
Several incorrect equations and expressions have been found in 
the above paper’. Some of these incorrect equations are corrected 
and presented in correspondence. There are still two errors which 
have not been corrected yet. These remaining errors are indicated 
and corrected in this comment. 
Error I: On p. 393 of the above paper’, a synthetic trace gener- 
ation algorithm is shown in pseudo-Pascal for the infinite memory 
model. There is a mismatch between the number of open and close 
parentheses in the following assignment statement of the algorithm. 
iiidpx : = rouncl((~-/(.-~’ )/o))(”(’-‘))) 
ID. Thiebaut, J. L. Wolf, and H. S. Stone, IEEE Trans. Comput., vol. 41, 
pp. 388-410, Apr. 1992. 
*D. Thiebant, J. L. Wolf, and H. S. Stone, “Corrigendum to ‘Synthetic trace 
for trace-driven simulation of cache memories,”’ IEEE Trans. Comput., vol. 
42, pp. 635-636, May 1993. 
Manuscript received April 13, 1993; revised August 6, 1993. 
The author is with the Department of Electrical and Computer Engineering, 
IEEE Log Number 9212757. 
Wayne State University, Detroit, MI 48202. 
0018-9340/94$04.00 0 1994 IEEE 


A Note on “A Systematic (12,8) Code for Correcting Single Errors and Detecting Adjacent Errors”

Introduction to Coding Theory.

Optimal codes for correcting single errors and detecting adjacent errors,’’

revised August 6,

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A Note on “A Systematic (12,8) Code for Correcting Single Errors and Detecting Adjacent Errors”

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