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Markers to construct DC free (d, k)
constrained balanced block codes using
Knuth’s inversion
H.C. Ferreira, J.H. Weber, C.H. Heymann and
K.A.S. Immink
In this reported work, Knuth’s balancing scheme, which was originally
developed for unconstrained binary codewords is adapted. Presented is
a simple method to balance the NRZ runlength constrained block codes
corresponding to (d, k ) constrained NRZI sequences. A short marker
violating the maximum runlength or k constraint is used to indicate
the balancing point for Knuth’s inversion. The marker requires fewer
overhead bits and less implementation complexity than indexing the
balancing point’s address by mapping it onto a (d, k) or runlength con-
strained prefix, such as when applying Knuth’s original scheme more
directly. The new code construction may be attractive for future mag-
netic and especially optical recording schemes. In fact the current
optical storage media, such as the CD, DVD and Blue Ray Disc, all
attempt to achieve some suppression of low frequency components
of the constrained codes, by exploiting a limited degree of freedom
within the set of candidate (d, k) words.
Introduction: Binary (d, k) constrained codes have been used since the
1960s in magnetic recording and are currently very prominent in optical
recording [1]. Here d denotes the minimum number of 0s and k the
maximum number of 0s between any two 1s in the NRZI representation
of the coded sequence. Hence there is a minimum runlength of d + 1
same symbols and maximum runlength of k + 1 same symbols in the
NRZ representation of the binary runlength constrained channel
sequence. By balancing the binary channel symbols {0, 1} as in [2]
and mapping them onto bipolar symbols {2 1, +1}, a balanced and
hence DC free code with suppressed low frequency components can
be constructed. Previously, investigations of DC free (d, k) constrained
codes focused mainly on short finite state machine codes for rotating
head magnetic tape recorders as in [3], or medium length block codes
as currently implemented in optical storage media [1]. Long block
codes as in [4] may suffer from large error propagation. However,
long block codes are attractive in order to more closely approach
channel capacity and hence increase storage density. There is very
little literature on constructing such long block codes. We now present
a simple method, requiring fewer overhead bits and also less implemen-
tation complexity than the more direct application of Knuth’s algorithm
in [5].
Code construction: We modify and apply Knuth’s balancing scheme
[2], which was originally developed for unconstrained binary code-
words. The main idea in Knuth’s construction is to convert the original
data sequence into a balanced sequence by inverting all symbols beyond
a certain balancing index. This index is represented in a (balanced)
prefix, which is also communicated. The receiver obtains the index
from the prefix and retrieves the original data sequence by inverting
all symbols beyond the balancing index. In the construction for balanced
constrained sequences which is presented in this Letter, an alternative to
the prefix method is proposed. Specifically, a short marker violating the
maximum runlength constraint is now used to indicate the balancing
point for Knuth’s inversion. Note that while the d constraint is still criti-
cal for recording density and reliability, the k constraint increased in
steps from k ¼ 3 in the 1960s when only simple tank oscillator elec-
tronic synchronisation circuitry was available, to k ¼ 7 in the 1970s
and later k ¼ 11. So far there has been little interest in further increasing
k, since this will only yield an insignificant increase in practical record-
ing density as can be shown by numerical evaluation of the information
theoretic channel capacity as in e.g. [1]. With state of the art phase
locked loops, occasional violations are tolerable. In fact, it can also be
expected that in future synchronisation technology may be further
improved. In this Letter we exploit occasional violations to create
markers.
We use prior art such as in [1] or [6]
to generate a high rate NRZI
(d, k ) constrained block code. Represent the NRZ (d + 1, k + 1)
runlength constrained channel sequence x with
x
1
...x
j
x
j+1
...x
n
x
i
[ {0, 1}(1)
where j is the balancing index. Consequently, the sequence can be
balanced by inverting the last n–j bits x
j+1
...x
n
. Knuth showed that
every binary word has at least one such balancing index j, where 1 ≤
j ≤ n. Let
x
i
be the binary complement of x
i
. We now insert a balanced
marker sequence M ¼ m
1
...m
z
of length z bits and transmit
x
i
...x
j
m
1
...m
z
x
j+1
...x
n
(2)
We want to extend the last run of the sequence x
1
...x
j
to create a
maximum runlength constraint violation, i.e. a run of k + 2 bits or
longer, without violating the minimum runlength of d + 1 channel
symbols anywhere. To this end, we propose the following marker. Let
y
u
denote a run of length
m
same symbols y and set
M = x
k+1
j
x
k+1
j
x
d+1
j+1
x
d+1
j+1
(3)
If j ¼ n, then we use the first bit of the next word as
x
j+1
. As a simple
example for exposition, the set of NRZ candidate markers for a (d, k) ¼
(1, 3) code is
{000011110011, 000011111100, 111100001100, 111100000011}
Note that M is balanced and that the overall length of the marker is z ¼
2k + 2d + 4. The first maximum runlength violation which includes x
j
,
ranges between k + 2 and 2k + 2 symbols. Note that if x
j
= x
j+1
,a
second maximum runlength violation, of length k + d + 2, occurs
inside the marker. A third runlength violation, of length at most k +
d + 2, may occur where the marker M merges with the inverted
portion of the sequence. However, the decoder only searches for the
first violation. When the first maximum runlength violation is detected
the decoder counts back k + 1 bits from the end of this violated run to
locate the start of the marker. It then removes the marker, inverts
x
j+1
...x
n
and uses prior art to decode and retrieve the data.
The proposed marker is a very simple one. More advanced markers
can be designed for specific purposes, at the expense of a higher com-
plexity. For example, constructions of variable-length markers with a
shorter average length are possible by taking into account the lengths
of the runs preceding and following the balancing index. Also, the
marker construction may be optimised to bring down the maximum run-
length violation. Note that for the proposed marker a run of length 2k +
2 can occur in the transmitted sequence, which may be considered as
being too long. By extending the set of candidate markers the
maximum value of this runlength violation may be set closer to k + 2.
However, such advanced markers are beyond the scope of this Letter,
where we focus on the introduction of the marker concept and the
simple design presented in (3).
–20
–15
–10
–5
0
5
10
0 0.1 0.2 0.3 0.4 0.5
PSD, dB
frequency f
pure MFM
proposed construction, 16 bit
proposed construction, 128 bit
Fig. 1 Comparison of power spectral densities between MFM (solid curve)
and proposed construction, with (d, k) ¼ (1, 3), for 16 bit (dashed curve)
and 128 bit (dotted curve) codewords as input to balancing
Results: With the above marker construction, the overhead is 2k + 2d +
4 bits per codeword. We compare this to the previous construction, pre-
sented in [5], where Knuth’s algorithm is directly applied to constrained
sequences, which requires a fixed length interfix of d + 1 bits as well as
a prefix, of length dependent on (d, k) and n, as tabled in [5]. Note that
the overhead of the marker construction does not depend on n.
Therefore, except for small values of n, the proposed method based
on the marker has less overhead as well as lower complexity. For
example, if (d, k) ¼ (1,3) the marker presented here represents less over-
head if the prefix in [5] is longer than (2k + 2d + 4)–(d + 1) ¼ 12 – 2 ¼
10 bits.
The power spectral densities for (d, k) ¼ (1, 3) coded sequences with
markers corresponding to the proposed construction are compared to the
(d, k) ¼ (1,3) MFM (Miller code, [1])inFigs. 1 and 2.InFig. 1, the
ELECTRONICS LETTERS 13th September 2012 Vol. 48 No. 19
influence of the codeword length is shown as well. In Fig. 2, it can be
seen how the low frequency components are suppressed.
–20
–15
–10
–5
0
5
10
10
–3
10
–2
10
–1
PSD, dB
frequency f
pure MFM
oroposed construction, 128 bit
Fig. 2 Low frequency performance comparison for MFM (solid curve) and
proposed construction (dotted curve), with (d, k) ¼ (l, 3), for 128 bit code-
words as input to balancing
Conclusion: We have presented an efficient construction for long
balanced and DC free (d, k) constrained block codes. In the context of
constrained codes, modifying Knuth’s original scheme by using
markers instead of the original Knuth’s prefix – which for (d, k) con-
straints furthermore mandates an additional interfix – may require con-
siderably fewer overhead symbols and lower implementation
complexity.
# The Institution of Engineering and Technology 2012
19 April 2012
doi: 10.1049/el.2012.1220
H.C. Ferreira and C.H. Heymann (Department of Electrical and
Electronic Engineering Science, University of Johannesburg, APK
Campus, Box 524, Auckland Park, 2006, South Africa)
E-mail: hcferreira@uj.ac.za
J.H. Weber (Delft University of Technology, The Netherlands)
K.A.S. Immink (Turing Machines, Rotterdam, The Netherlands)
References
1 Immink, K.A.S.: ‘Codes for mass data storage systems’ (Shannon
Foundation Publishers, Eindhoven, The Netherlands, 2004, 2nd edn),
ISBN 90-74249-27-2
2 Knuth, D.E.: ‘Efficient balanced codes’, IEEE Trans. Inf. Theory, 1986,
IT-32, (1), pp. 51 – 53
3 Ferreira, H.C.: ‘On dc free magnetic recording codes generated by finite
state machines’, IEEE Trans. Magn., 1983, MAG-19, (6),
pp. 2691 –2693
4 Braun, V., and Immink, K.A.S.: ‘An enumerative coding technique for
DC-free runlength-limited sequences’, IEEE Trans Commun., 2000,
48, (12), pp. 2024 – 2031
5 Immink, K.A.S., Weber, J.H., and Ferreira, H.C.: ‘Balanced runlength
limited codes using Knuth’s algorithm’. Proc. of IEEE Int. Symp. on
Information Theory, St Petersburg, Russia, July– August 2011,
pp. 282–285
6 Adler, R., Coppersmith, D., and Hassner, M.: ‘Algorithms for sliding
block codes – an application of symbolic dynamics to information
theory’, IEEE Trans. Inf. Theory, 1983, 29, (1), pp. 5–22
ELECTRONICS LETTERS 13th September 2012 Vol. 48 No. 19
