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ISIT
1997,
Ulm, Germany,
June
29

July
4
On the Construction
of
Constrained Codes employing Sequence
Replacement Techniques
A.J.
van Wijngaarden and
K.A.
Schouhamer Immink
*
Institute for Experimental Mathematics, Digital Communications Group, Ellernstr.
29, 45326
Essen, Germany
Philips Research Laboratories, Prof. Holstlaan
4, 5656
AA
Eindhoven, The Netherlands
Abstract

General construction methods
of
pre
Ax
synchronized
codes
and
runlength limited
codes
are
presented, which make use
of
socalled
sequence
replacement techniques. These techniques provide
a
simple and efficient conversion
of
data
words into
codewords
of
a
constrained blockcode, where sub
sequences violating the imposed constraints
are
re
placed
by
encoded
information
to
indicate their
rel
ative positions in the
data
word. Several construc
tions
are
proposed
for
constrained
codes
with low er
ror propagation, and for variable length constrained
codes.
The
coding algorithms have
a
low computa
tional and hardware complexity. The
rate
of
the con
structed
codes
approaches the theoretical maximum.
It
is
feasible
to
apply these high
rate
constrained block
codes
in communication and recording systems.
and prefix
P
of length
k
=
[log,(n)
+
11
can be constructed.
The redundancy
is
minimum for many values of
n.
We can construct RLL codes with
a
maximum runlength
constraint using the replacement technique. These RLLcodes
correspond to
(0,
k)
codes
[3],
which consist of
a
set
of
binary
sequences with the property that at most
k
zeroes occur be
tween two consecutive ones. High rate constrained
(0,
k)
codes
can be construct.ed for relevant values of
k
by converting data
words of length
(n

l)
into constrained words of length
n.
Table
I
lists the maximum word length
nmax
for which
a
rate
n,
(0,
k)
code exists, and the maximum word length
nconJtr
for
which a rate
+,
(0,
k)
code can be constructed. The
same parameters
are
given for concatenatable
(0,
k)
codes.
n1
TABLE
I
MAXIMUM
LENGTH
n
FOR
WHICH
A
RATE
e,
(0,
k)
CODE
AND
A
CONCATENATABLE
(0,
k)
CODE CAN BE CONSTRUCTED.
I.
INTRODUCTION
We consider the construction of constrained blockcodes,
which have the property that
a
specific subsequence does not
occur in the codewords. These codes
are
in particular useful
for the realization of frame synchronization in communica
tion systems by using prefixsynchronized (PS) codes
[I, 21,
or
to fulfil the channelinput constraints
of
magnetic and
optical recording systems by using runlengthlimited (RLL)
codes
[3, 41.
Several constructions
of
PS
codes have been
pro
posed
in
the past, among which
a
systematic code
[l],
and an
enumerative code
[2,
51.
For
practical applications, the sys
tematic code
is
too inefficient, and the implementation
of
enu
merative codes
is
quite complex.
For
this reason, lowcomplex
bitstuffing procedures are used, e.g., in data networks
[6].
In
recording systems, RLLcodes have thus far been constructed
using small codes employing lookup tables
or
combinatorial
circuitry. The new techniques offer promising alternatives.
The basic idea is to replace subsequences that violate the
imposed constraints by information about their position. The
number of forbidden subsequences and their location
is
en
coded and stored in the codeword. This enables the decoder
to reconstruct the violations
at
the indicated positions. The
codeword length
is
constant for given data word length.
11.
CONSTRAINED CODE C.ONSTRUCTIONS
A
PScode is
a
set of codewords with the property that each
codeword starts with
a
prefix
P
of
length
k,
followed by
a
sequence
2
of
length
m.
Prefix
P
marks the boundaries of
each codeword, and should therefore not occur at any other
position in the codeword. If
P
is
of the form
1..
.lo,
this
condition is fulfilled if
P
does not occur anywhere in sequence
Z
[5].
The sequence replacement technique is used
to
convert
a
data sequence of length
m

1
into
a
constrained sequence
of length
m
where
P
does not occur. We will show, that for
given symbol set size
CY
a
PS
code of word length
n
=
k
+
m
(0,
k)
code
nmax
nconstr
rate concat.
(0,
k)
code
nmax
nconstr
rate
3
4
5
6
7
8
9
10

21
43
88
177
355
710
1420
2840
15
32
65
130
259
516
1029
2054
0.9333
0.9688
0.9846
0.9923
0.9962
0.9980
0.9990
0.9995
12
31
67
148
310
649
1327
2715
5
14
39
88
201
426
907
1868
0.8000
0.9286
0.9744
0.9944
0.9950
0.9977
0.9989
Table
I
shows that the constructed
(0,k)
codes have
a
very
high rate. The complexity of the encoding and decoding al
gorithms is low and nearly independent of the word length.
It
will be shown that the new constructions have lower er
ror
propagation than the bitstuffing and enumerative coding
technique.
A
combined constrained coding and error control
scheme is proposed
to
suppress error propagation. This is
an
essential requirement
for
the actual application
of
the con
structed high rate, constrained codes.
REFERENCES
[l]
E.
N.
Gilbert, “Synchronization
of
binary messages,”
IRE
Trans. Inform. Theory,
pp. 470477, Sept. 1960.
[2]
W.
H.
Kautz, “Fibonacci codes for synchronization control,”
IEEE
Trans. Inform. Theory,
vol.
11,
pp. 284292,
Apr.
1965.
[3]
K.
A.
S.
Immink,
Coding Techniques
for
Digital Recorders.
Prentice Hall, Englewood Cliffs, New Jersey, 1991.
[4]
C.
D. Mee and
E.
D. Daniel,
Magnetic Recording.
McGrawHill,
New
York,
1987.
[5]
H.
Morita,
A.
J.
van Wijngaarden, and
A.
J.
H.
Vinck, “On
the construction
of
maximal prefixsynchronized codes,”
IEEE
Trans. Inform. Theory,
vol. 42, pp. 21582166, Nov. 1996.
[6]
D.
Bertsekas and
R.
G. Gallager,
Data Networks.
Prentice
Hall,
second ed.. 1987.
0780339568/97/$10.00 01997
IEEE
144
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