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complexity has a low order of n log n or even a linear order n if incomplete hard
decision decoding is concerned.
This combination of powerful end decoding and fast intermediate processing
outperforms all other nonexponential algorithms known for RM codes. In
particular, it turns out that the algorithms of [56-58] substantially outperform
the two conventional techniques of bounded distance decoding and majority
decoding. Namely, new recursive techniques
increase 1ndtimes the number of errors d/2 corrected by bounded-distance
decoding (for most error patterns);
double the number of errors (d 1nd)/4 corrected (with high probability) by
conventional majority decoding; for the moderate lengths of 128 to 512,
these techniques were enhanced in [4] with list decoding, which
gives near-optimum performance (within 0.25 dB of ML decoding) with
feasible complexity;
outperforms all other decoding algorithms currently known for all existing
codes.
Some results are given below in Figure 6.14a and b for different RM codes of
lengths 256 and 512, respectively. An important conclusion is that RM codes yield
near-optimum performance and fast processing with complexity of order n log n.
Then in Figure 6.15a and b, also the results for longer blocks obtained by
concatenating different RM codes are plotted. In summary, these data illustrate
that concatenated RM codes combine fast feasible processing and very low SNR
of 2 to 3 dB on the Gaussian channels for various block lengths of 256 to 4000 bits.
These algorithms could be further enhanced in the future to correct specific (non-
Gaussian) error models that pertain to magnetic recording channels.
6.4. CONCLUSIONS
This chapter outlines multidisciplinary research to develop a multilevel mode of
next-generation magnetic recording, a promising and challenging solution to
increasing the capacity of electronic and computer devices. The study is conducted
from the system perspective. It is anticipated that through marrying the expertise in
magnetic recording and state-of-the-art data encoding, the proposed multilevel
magnetic recording approach could be applied to any alternative magnetic data
storage technology to further increase its data capacity by several factors. The key
to the multilevel system is exploitation of the recently proposed concept of three-
dimensional media optionally with patterned soft magnetic underlayer and inter-
layers. Among the unique properties are the substantially increased (compared
with conventional technologies) recording and sensitivity fields used for writing
and reading information, respectively. This field increase results in adequate signal
to noise ratio (SNR) to accommodate over 40 signal levels with 5 dB difference
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