Near maximum likelihood soft-decision decoding of a particular class of rate-1/2 systematic linear block codes

Abstract

Presented is a soft-decision decoding algorithm for a particular class of rate-1/2 systematic linear block codes. The proposed algorithm performs successive re-encoding of both the data and parity bits, to produce a list of codewords among which the most likely candidate is chosen. Simulation results show that close to optimal performance can be obtained at reasonable complexity. They validate the potential of the proposed algorithm as a practical approach for soft-decision decoding.

Full text

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Near maximum likelihood soft-decision decoding of a
particular class of rate-1/2 systematic linear block codes

P. Adde*, C. Jégo** and R. Le Bidan*

This paper presents a soft-decision decoding algorithm for a particular class of rate-
1/2 systematic linear block codes. The proposed algorithm performs successive re-
encoding of both the data and parity bits, to produce a list of codewords among
which the most likely candidate is chosen. Simulation results show that close to opti-
mal performance can be obtained at reasonable complexity. They validate the poten-
tial of the proposed algorithm as a practical approach for soft-decision decoding.

Introduction: This paper focuses on Soft-Decision Decoding (SDD) of a particular
class of rate-1/2 (n=2k,k) systematic binary linear block codes characterized by the
property that their generator matrix G in canonical form can be written as G = [Ik, P],
where Ik is the k×k identity matrix and P is constrained to be a k×k invertible binary
matrix. Note that the considered code family includes self-dual codes as a particular
case since the latter satisfy P-1 = Pt [1]. Many interesting block codes falls into this
category, including the (24,12,8) extended Golay code, self-dual Reed-Muller codes,
extended quadratic residue codes, or the more recent Cortex codes [2]. The Cortex
construction offers a broad spectrum of possibilities for the problem of constructing
short linear block codes with good minimum distance properties and low-complexity
soft-decision decoders.
Previous work: Maximum-likelihood (ML) soft-decision decoding is known to offer
the best decoding performance in terms of block error probability. However it is com-
putationally intractable for most codes of practical interest. Hence sub-optimal SDD
algorithms usually have to be considered. Belief Propagation (BP) is a soft-input soft-
output decoding algorithm relying on the exchange of soft information along the
edges of a graph defined by the parity-check matrix H of the code [3]. BP is known to
closely approximate the performance of optimal Maximum A Posteriori (MAP) decod-
hal-00573239, version 1 - 3 Mar 2011
Author manuscript, published in "Electronics Letters 47, 4 (2011) 259 -260"
DOI : 10.1049/el.2010.7857

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ing at reduced complexity for codes with sparse parity-check matrices. On the other
hand, BP performs poorly with most block codes since the latter usually have high-
density parity-check matrices, and thus graphs with many short cycles. A very gen-
eral approach to SDD builds a list of candidate codewords and selects the codeword
at minimum Euclidean distance from the received word in this list. Various strategies
have been proposed to generate the list. A first category of algorithms performs suc-
cessive re-encoding of different information sets [4-7]. We recall that an information
set is any set of k linearly independent coordinates in an (n,k) linear block code.
These k positions uniquely determine the codeword, and thus can be used for sys-
tematic encoding. In another family of algorithms, the list is constructed by applying
candidate error patterns to the received word and by running a hard-decision de-
coder on the resulting sequences [8,9]. In both cases, reliability information available
at the matched-filter output can be taken into account during the generation of candi-
date codewords to significantly reduce the list size (and thus the decoding complex-
ity) at a given performance level.
Proposed algorithm: The proposed algorithm falls into the class of reliability-based
information-set SDD algorithms. It takes advantage of the code properties to further
simplify the decoding process. Assuming Binary Phase-Shift Keying (BPSK) trans-
mission over an Additive White Gaussian Noise (AWGN) channel, the algorithm op-
erates as follows. Let r = (r0, …, rn-1) denote the received sequence at the matched-
filter output, and z = (z0, …, zn-1) the corresponding hard-decision sequence (zi = 1 if ri
> 0, and 0 otherwise). The vector z is split into two parts z = (zs, zp) by separating the
k systematic and k parity coordinates, respectively. A series of Nep error patterns is
applied to the k systematic coordinates zs which are then re-encoded using the ca-
nonical generator matrix Gs = [Ik, P], to produce a first list of Nep candidate code-
words. A second list of Nep candidates is obtained by repeating the procedure in par-
allel on the k parity coordinates zp. Re-encoding of the parity set zp is realized using
the generator matrix Gp = P-1 × Gs = [P-1, Ik]. By construction, the matrices Gs and Gp
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are equivalent up to elementary row operations and thus generate the same code.
The final decision is given by the candidate codeword at minimum Euclidean dis-
tance from the observation r.
Comparison with other SDD algorithms: Compared to the more general informa-
tion-set SDD algorithms described in [5][6], the proposed algorithm is computationally
much simpler since it does not need to identify the most reliable information set for
each received vector, thereby avoiding a costly Gaussian elimination process. In-
stead, our algorithm takes advantage of the code symmetry to operate on two, fixed
information sets, for which encoding functions are easily derived. Further, unlike
[8][9], no hard-decision decoder is required. Hence this algorithm can be applied to
codes for which no algebraic decoder exists. The error patterns generation strategy
also differs from the one used in [5] and [6]. In [5] (see also [10]), error patterns are
generated in order of increasing Euclidean weight, whereas order-i ordered-statistics
decoding (OSD) in [6] exhaustively tests all combinations of i errors or less in the k
most reliable information coordinates. In the proposed algorithm, similarly to the
Chase-2 algorithm [9] (see also [7]), the Nep = 2p candidate error patterns are con-
structed by testing all configurations of errors confined in the p least reliable positions
in each of the considered information set (zs or zp).
Simulation results: Figure 1 shows the bit-error-rate (BER) performance of the pro-
posed algorithm for the (24,12,8) extended Golay code. For reference purpose, the
performance of ML and OSD decoding are also shown. Order-1 and order-2 OSD
involve 13 error patterns and 79 error patterns, respectively. We observe that 2×16
error patterns are sufficient to reach near-ML performance with the proposed algo-
rithm. Order-1 OSD performs slightly better with less error patterns, but this comes at
a much higher computational cost, due to the Gaussian elimination process. Order-2
OSD achieves ML performance but with 79 error patterns.
The BER performance of several rate-1/2 block codes of length n=8 up to 48 under
SDD with the proposed algorithm are shown in Figure 2. The considered codes in-
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clude classical algebraic constructions such as the (8,4,4) extended Hamming code,
the (24,12,8) extended Golay code or the (48,24,12) extended quadratic residue
code. The (16,8,5) code was obtained using the Cortex construction. First introduced
in [2], Cortex codes are a family of rate-1/2 self-dual systematic linear block codes
with good minimum distance properties. The Cortex construction combines a short
base code C with a finite sequence of permutations to produce the k parity bits. The
reader is referred to [2][11] for a detailed description of the encoding process. The
number of error patterns used in the simulations is specified in Table 1. It was opti-
mized in order to obtain performance within 0.05-0.25 dB from ML decoding for each
code. For comparison purpose, we also give the number of error patterns required by
order-dmin/4 OSD which was shown in [6] to practically achieve ML for block codes
of length n ≤ 128. These results show that the proposed algorithm is able to achieve
near-ML performance at reduced complexity. Indeed, fewer error patterns are neces-
sary and no Gaussian elimination operation (costly in terms of latency and hardware
resources) is employed.
Conclusion: A soft-decision decoding algorithm has been introduced for a specific
class of rate-1/2 systematic linear block codes which includes in particular the family
of self-dual codes. The algorithm performs successive re-encoding of two fixed in-
formation sets to produce a list of candidate codewords. Simulation results show that,
by taking advantage of the code structure, the proposed algorithm realizes a good
trade-off between performance and complexity, thereby providing an attractive alter-
native to other classical soft-decision decoding algorithms.
References

[1] R. E. Blahut, Algebraic codes for data transmission. Cambridge University Press,
2003.
[2] J. C. Carlach and C. Vervoux, "A new family of block turbo-codes," in 13th Int.
Symp. on Applied Algebra, Algebraic Alg. and Error-Correcting Codes, Honolulu,
USA, November 1999.
[3] F. R. Kschischang, B. J. Frey, H.-A. Loeliger, "Factor Graphs and the Sum-
Product Algorithm," IEEE Trans. Inform. Theory, vol. 47, no. 2, Feb. 2001, pp. 498-
519.
hal-00573239, version 1 - 3 Mar 2011

Page 5

[4] J. MacWilliams, “Permutation decoding of systematic codes,” Bell Sys Tech. J.
Jan. 1964, pp. 485--505
[5] B. G. Dorsch, “A decoding algorithm for binary block codes and J-ary output
channels”, IEEE Trans. Inform.Theory, vol.20, n° 3, pp.391-394, May 1974
[6] M. P. C. Fossorier and S. Lin, “Soft-decision decoding of linear block codes based
on ordered statistics”, IEEE Trans. Inform.Theory, vol.41, n° 5, pp. 1379-1396,
Sept.1995
[7] M. Lalam, K. Amis and D. Leroux, “Sliding window encoding for Reed-Solomon
code decoding,” in Proc. 4th Int. Symp. Turbo Codes, Munich, Germany, April 3-7,
2006
[8] G. D. Forney Jr, “Generalized minimum distance decoding”, IEEE Trans. Inform.
Theory, vol. 12, n° 2, pp. 125—131, Apr. 1966
[9] D. Chase, “A class of algorithms for decoding block codes with channel meas-
urement information,” IEEE Trans. Inform. Theory, vol IT-18, Jan. 1972, pp 170-182
[10] A. Valembois & M. P. C. Fossorier, “An improved method to compute lists of bi-
nary vectors that optimize a given weight function with application to soft-decision
decoding”, IEEE Commun. Lett., vol.5, n° 11, pp.456--458, Nov. 2001
[11] J. E. Perez Chamorro, F. Seguin, C. Lahuec, M. Jezequel, G. Le Mestre, "De-
coding a family of dense codes using the sum-product algorithm and subthreshold
PMOS, " in Proc. IEEE Int. Symp. Circuits and Systems ISCAS’09, Taiwan, 24-27
May 2009, Taipei, Taiwan, pp. 2685-2688.



Authors’ affiliations:
* Institut TELECOM; TELECOM Bretagne; CNRS Lab-STICC UMR 3192
** CNRS IMS, UMR 5218, 351 Cours de la Libération, 33405 TALENCE,
FRANCE


Figure captions:

Figure 1: BER performance of soft decoding of the (24,12,8) Golay code.

Figure 2: Soft-decision decoding performance of different rate-1/2 block codes of
length n=8 up to 48 with the proposed decoding algorithm


Table

Table 1: Number of error patterns required by the proposed solution and by OSD for
different codes (n,n/2) block codes (n=8 up to 48).









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Figure 1
0
1,00E-10
1,00E-09
1,00E-08
1,00E-07
1,00E-06
1,00E-05
1,00E-04
1,00E-03
1,00E-02
1,00E-01
1,00E+00
12345678
Eb/N0 in dB
BER
ML
proposed 2*16v
proposed 2*8v
proposed 2*4v
order-1 OSD 13v
order-2 OSD 79v



Figure 2
0
1,00E-12
1,00E-11
1,00E-10
1,00E-09
1,00E-08
1,00E-07
1,00E-06
1,00E-05
1,00E-04
1,00E-03
1,00E-02
1,00E-01
1,00E+00
12345678
Eb/N0 in dB
BER
n=8 2*2v
n=16 2*4v
n=24 2*16v
n=48 2*512v



Table 1
Code (8,4,4) (16,8,5) (24,12,8) (48,24,12)
2×2 2×4
5 9
Proposed algorithm
Order-   dmin/4    OSD

2×16
79
2×512
2325
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