In this section, some simulated performance of two SEOLD algorithms (SEOLD-I and SEOLD-II) are presented and compared with that of other well known RS decoding algorithms. A standard binary input AWGN channel is assumed over which the BPSK modulated codewords are transmitted. We model the receive matched filter output as the sum of a±1−valued sequence and Gaussian sequence with zero-mean i.i.d. components.
The average performance bounds on the ML error probability of RS codes over an AWGN channel developed in [25] are used as the performance lower limits.
Due to the complexity and the decoding delay considerations, the SEOLD algorithms will not terminate until convergence is assured. Instead, we limit our decoding procedure to T iterations in all simulations.
Fig. 3.4 shows the codeword error rate (CER) performance of the (15,11) RS code over an AWGN channel. HDD-BM refers to the performance of a hard decision bounded minimum distance decoder such as the BM algorithm. GMD and KV refer to the per-formance of the GMD algorithm proposed by Forney and the algebraic soft decision decoding algorithm proposed by Koetter and Vardy, respectively. Note that the KV algorithm concerned here is infinite interpolation costs, i.e., the complexity is also infi-nite. For both SEOLD-I and SEOLD-II, the size of the sample set N and the size of the elite set E at every iteration are set to be 20 and 6, respectively. After 10 iterations, SEOLD-I has about 0.5 dB and 0.3 dB coding gain over GMD and KV at a CER of 10−5,
1 2 3 4 5 6 7 8 10-6
1x10-5 1x10-4 10-3 10-2 10-1 100
Codeword Error Rate
Eb/N
0 (dB) HDD-BM
GMD KV
SEOLD-I ( N =20) SEOLD-II ( N =20) ML
Figure 3.4: Codeword error probability performance of the (15,11) Reed-Solomon code;
10 iterations.
respectively. On the other hand, SEOLD-II outperforms all the previous algorithms with a performance gain of about 1.2 dB and 1.0 dB over GMD and KV at a CER of 10−5.
In Fig. 3.5, SEOLD-I has a near KV performance when the N = 100 and E = 10 after 10 iterations. At the same condition, SEOLD-II still outperforms the other algorithms with reasonable complexity. The SEOLD-II has about 0.6 dB and 1.0 dB coding gain over KV where N is equal to 100 and 500, respectively. In conclusion, the proposed decoding algorithms are capable of offering good performance with modest complexity for short high rate RS codes. Its performance can be further improved by increasing the sample size N and/or the maximum iteration number T at the cost of increased decoding complexity.
1 2 3 4 5 6 7 8 10-6
1x10-5 1x10-4 10-3 10-2 10-1 100
Codeword Error Rate
Eb/N
0 (dB) HDD-BM
GMD KV
SEOLD-I ( N =100) SEOLD-II ( N =100) SEOLD-II ( N =500) ML
Figure 3.5: Codeword error probability performance of the (31,25) Reed-Solomon code;
10 iterations.
Chapter 4
Stochastic List Decoding of Linear Block Codes
In the previous chapter, we focus on decoding RS codes of short to medium length. The MDS character of RS codes is exploited to reduce the complexity of locating near-by codewords. Such an approach cannot be applied to general linear codes. We thus present a new decoding method which is valid for arbitrary linear codes but is more effective for codes with small girth.
4.1 Preliminary
Let C be a binary (N, K) linear block code with minimum distance dmin and M × N parity-check matrix H. As the rows of H may be dependent, we have M > N− K. Let I = {1, · · · , N} and J = {1, · · · , M} be the sets of column indices and row indices of H, respectively. We denote the set of bits n that participate in check m by N (m) = {n : Hmn = 1}. Similarly, we define the set of checks in which bit n participates as M(n) = {j : Hmn = 1}. We denote a set N (m) with bit n excluded by N (m)\n, and a set M(n) with parity check m excluded by M(n)\m. The cardinality of N (m) and M(n) are denoted by |N (m)| and |M(n)|, respectively. Let en be a 1× N elementary vector with 1 at position n and 0 at other entries.
An 1× N vector c is a codeword of C if and only if cHT = 0 where HT is the
transpose of H and 0 is a 1× M zero vector. For each row hm of H, m∈ J, let
Using the binary phase-shift-keying (BPSK) signal, the transmitter maps a codeword c into the bipolar vector
Ψ(c) = x = (x1,· · · , xN), xn = Ψ(cn) = (−1)cn (4.3)
and sends it over an additive white Gaussian noise (AWGN) channel with zero mean and power spectral density N0/2 W/Hz. The received sequence at the output of the matched filter is given by y = (y1,· · · , yN), where yn= xn+wnand wn’s are statistically independent Gaussian random variables with zero mean and variance N0/2.
Let z = (z1,· · · , zN) be the hard decision version of the received sequence y, i.e.,
zn =
( 0, yn > 0
1, otherwise (4.4)
For m∈ J, we define σm as the result of check sum-m based on the hard-decision vector z: of the log-likelihood ratio (LLR) associated with the corresponding hard-limited bit zn
Ln = logP (cn= 0| y)
P (cn= 1| y). (4.6)
We also denote L = (L1,· · · , LN) as the LLR vector of the received word.
Let λm be the reliability of check sum m which is defined as
λm = min
n∈N (m)γn (4.7)
Then we first sort {λm : m ∈ J} and let m1, m2,· · · , mM denote the position of the check sums in terms of descending order of {λm : m∈ J}, i.e., the check sum m1 is the most reliable and mM is the least reliable.
Define qn = P (zn 6= cn|y) as the a posteriori probability that bit n is in error based on y. Then we have the following lemmas.
Lemma 4.1 For the AWGN channel model considered, the probability qn can be ex-pressed as
qn = 1
1 + eγn (4.8)
Proof :
See Appendix A.
Lemma 4.2 The probability that for check sum m ∈ M(n), the sum of all bits n0 ∈ N (m)\n mismatches the transmitted bit n0, say rmn, is
rmn = 1 2
1− Y
n0∈N (m)\n
(1− 2qn0)
. (4.9)
Proof :
See [26].
Note that rmn represents the probability of having an odd number of errors N (m)\n.
Define ˜qn as the a posteriori probability that bit n is in error based on the results of the check sums intersecting in position-n. Then we obtain the following useful theorem.
Theorem 4.1 Given the received word y and the syndrome set Σn ≡ {σm : m∈ M(n)}, the logarithm of the bit correctness probability ratio for bit n, say ξn, is
ξn = log
In this section, we introduce a single-run sequential bit-flipping (SBF) algorithm for transforming z into a valid codeword. This procedure has a special constraint about the parity-check matrix H that H has to be a systematic form. First of all, consider the rows of the parity-check matrix H are linearly independent, i.e., M = N − K.
Using appropriate row operations, H can be transformed into a systematic form, say H = [I˜ MP], where IM is an M × M identity matrix and P is an M × (N − M) binary matrix. Note that both H and ˜H are the null space of C, hence we can decode the received word by using ˜H instead of H.
However, it is impossible to have this transformation when the rows of H are linearly dependent, i.e., M > N − K. Fortunately, we can remove M − N + K rows of H which can be represented by the linear combination of the remain rows to get a (N − K) × N sub-matrix H0 where H0 has its systematic form ˜H0.
Example 4.1 Consider the following parity check matrix:
H =