Notas a los estados financieros (continuación)
_________ _________ 19. Transacciones con Pesquera Diamante S.A
22. Objetivos y políticas de gestión de riesgos financieros
decoding strategy locates the errors jointly in all RS codewords instead of locating them independently in the several words. Up to t errors can be located uniquely, in many cases even if ? is larger than half the minimum distance of the RS code with the largest dimension. In 1999, Guruswami and Sudan (GS) [11] proposed a new method to improve the er- ror correcting capability of RS codes beyond their traditional capability (half the minimum distance). Later, Parvaresh and Vardy [77] proposed multivariate interpolation decoding of RS codes based on GS algorithm and showed that if errors happen simultaneously for multiple codewords of an RS code, errors beyond GS algorithm can be corrected.
In this chapter, we derive and analyze an algorithm for collaborative decoding of het- erogeneous 1RS codes in the presence of burst errors based on multivariate interpolation decoding of RS codes [77]. Similar to GS algorithm, our method has two steps: interpo- lation and factorization. We find the error correction capability of the proposed algorithm and show that it is larger than the decoding radius of GS algorithm for the RS code with the largest dimension. Then, we analyze the performance of concatenated codes using 1RS codes as their outer codes. We derive upper and lower bounds for the word error probability
of GC codes over AWGN channel with BPSK modulation for both cases of independent
and collaborative decoding of the outer 1RS codes. We will show that using collaborative decoding, the word error probability is better compared to the case of independent decod-
ing.
The rest of this chapter is organized as follows. In section 5.1, interleaved Reed- Solomon codes are introduced. The proposed collaborative decoding of 1RS codes based on GS algorithm [11] is explained in detail in section 5.2 including the interpolation step and the factorization step. The error correction capability of the proposed method is also
derived in this section. In section 5.3, the use of 1RS codes in concatenated codes is dis-
cussed and the performance of GC codes over AWGN channel with BPSK modulation is considered. Also, lower and upper bounds on the performance of GC codes are derived. Numerical results and discussions are presented in section 5.4. Finally, conclusions are
given in section 5.5.
5.1
Interleaved Reed-Solomon Codes
As discussed in Chapter one, an RS code of length N and dimension K over the Galois
field GF(q) with support set D = {??, x<¿, ···, xn} C GF(q) is defined as
C9(N, K)=
(5.1)
{(f(xi),f(x2),..,fM)\xi,X2,...,XN e DJ(X) G Fq[X],degf(X) < K}
where Fq[X] is the ring of polynomials over the Galois field GF(q) in a variable X. RS
codes are maximum distance separable (MDS) and therefore from any set of K correct symbols, an RS codeword can uniquely be reconstructed. An interleaved Reed-Solomon code is now obtained by taking M Reed-Solomon codes over the same Galois field GF(q) and grouping them row-wise into a matrix. The codewords of an 1RS code are matrices
whose rows are the codewords of the Reed-Solomon codes. We denote these M Reed-
Solomon codes by RS^, RS®, ..., ÄS(M) where RS® = RS(N, Ku d¡), i = 1, 2, ..., M
and di = N - Ki + 1. Now, an interleaved RS code denoted by IRS(N, K1, K2, ..., KM) is defined as
I / ci \
C2IRS(N,K1,K2,...,KM) = {
[\cM J
,de RS®, i el,
M > - (5.2)If all the M Reed-Solomon codes are equivalent, i.e., RSW = RS^ = ... = RS^M\
the 1RS code is called homogeneous. Otherwise, we say that the 1RS code is heterogeneous.M
Information Symbols Party Symbols
ja m K2 K3
U
Km Borst Error5.2
Collaborative Interpolation Decoding of 1RS Codes
The basis of GS decoding algorithm [11] has been explained in Chapter 2, Section 2.3. In this section, we use the basis of GS algorithm and introduce collaborative decoding of 1RS codes assuming burst errors. In this method instead of decoding each RS codeword independently, we try to decode all the codewords simultaneously. In the end, we show that we can increase the error correction capability of GS algorithm using this method of
decoding.
Suppose that one codeword (c1, c2, ..., cM)T of an IRS(N, K1, K2, ..., KM) code,
corresponding to evaluations of the polynomials /1PO, P[X), —, fM(X) over GF(q)
with degrees less than K1, K2, ..., Km respectively, is transmitted over a hard decision channel and received as
/yl\
\yM J
/M
/ ei \
+V
cM J
\eM j
(5.3)where c\ y% and e\ i = 1, ..., M are vectors with N elements (symbols). In the case of
1RS codes, one codeword which is a matrix is transmitted across the channel column-by-
column. In the presence of burst errors in the channel, we can assume there are at most t synchronized errors meaning the error matrix has at most t nonzero columns. In this
case, the M Reed-Solomon codewords may have erroneous symbols at the same positions (columns).
Each codeword can be decoded separately at the decoder using the GS algorithm. From Chapter 2, Section 2.3, the asymptotic (m —>¦ oo) error correcting capability of the
GS decoder for an (N, Ki) RS code is U = N(I - ^/(K1 - I) /N) . Therefore, if each
codeword of the 1RS code is decoded independently using GS algorithm, the error correct-ing capability for the 1RS code is
where K — max {Ki, i = 1,2, ..., M}. However, since errors happen at the same posi-
tions, a collaborative decoding strategy can be applied to decode all the codewords at the same time. This may allow for correcting up to t synchronized errors with t > tg.
For collaborative decoding, we follow the basis of GS algorithm. First, we show
each of the received vectors y1, i = 1, 2, ...,M with {y\,y2, ---,2/V)- Given the support set
D = [X1, X2, .--,Xn] of GF(q), we consider the following set of points P in an (M + I)-
dimensional space:P = {(xi,y\,yì, ¦¦-,yìl),{.x2,yl,yl,---,y¥),---AxN,y1N,y2N,---,y%)} ¦
(5.5)
The collaborative decoding algorithm has two major steps (interpolation and factorization)
that will be explained in the following.
5.2.1
Interpolation step
In this step, given the point set P in GF(q)M and a positive integer m, we try to compute a
nontrivial (M + Invariate polynomial Qp(X, Y1, Y2, ..., YM) of minimal (1, K1 -1,K2-
1, ...,KM — 1) -weighted degree over GF(q) that passes through all the points in P with
multiplicity at least m.
The weighted degree of a polynomial can be defined as the weighted degree of its leading monomial. The (1, ^T1 — 1,K2 — 1, ..., Km — l)-weighed degree of the monomial