On the Probability of Undetected Error for Over ... - IEEE Xplore

2006 IEEE Information Theory Workshop, Punta del Este, Uruguay, March 13-17, 2006
On the Probability of Undetected Error for
Over-Extended Reed-Solomon Codes
Junsheng Han and Paul H. Siegel
Patrick Lee
Center for Magnetic Recording Research
Western Digital Corporation
University of California, San Diego
20511 Lake Forest Drive
La Jolla, California 92093-0401 Lake Forest, California 92630
Email: han@cts.ucsd.edu, psiegel@ucsd.edu
Email: patrick.lee@wdc.com
Abstract-We derive upper and lower bounds on the weight
of p, as follows:
distribution of Over-Extended Reed-Solomon (OERS) codes.
Using these bounds, we obtain tight upper and lower bounds n
on the probability of undetected error for OERS codes on q-ary Pud (P) Ai P (1 _ p)n-i (1)
symmetric channels. q -1
where Ai is the number of codewords with Hamming weight i.
I. INTRODUCTION Equation (1) relates the probability of undetected error directly
to the weight distribution of the code.
the received symbols appear to be uni-
In some applications, error correcting codes have been used When p = (q -1) lq,
as pure error detection codes. In particular, Reed-Solomon formly distributed no matter which codeword was transmitted.
(RS) codes have been used for error detection in some disk Therefore, undetected error occurs when the received word is
drives since the 1990's because they have excellent error detec- any codeword except the one sent and each such codeword
tion capabilities and do not exhibit the undesirable behavior appears with probability q-n. Since there are qk -1 such
characteristic of certain shortened binary cyclic redundancy incorrect codewords, we have
check (CRC) codes [1]. A further example is the USB interface
standard [2], which specifies the use of a Hamming code for p q- k 1-(-k)
error detection. Pud (q - 1)q 0).
from the one transmitted, in which case the errors will not Following [5], [9], [6], we call a code good if Pud(p) <
be detected. The probability of such an event is known as the q-(nk)for all 0 < p < q-1 and proper if Pud(p) is
probability of undetected error, and is denoted by Pud. monotonic in p for 0 < p < q-1 (Some authors have used
q
Consider an (ni,k) linear block code over F ,transmitted q-(nk) - qf as the goodness threshold. See [10].)Proper
over a q-ary symmetric channel, where each transmitted sym- codes are necessarily good, but not vice versa. Study of
bol is received correctly with probability 1 -p, and as any properness and goodness of certain classes of codes can be
of the other q -1 symbols with equal probability pj(q -1). found in [4], [5], [9], [6], [7]. Note that for the ensemble of
Clearly, for this channel, Pud can be calculated as a function all (ni, k) linear block codes over Fq, it is known [10] that the
1-4244-0036-8/06/$20.00 ©C2006 IEEE 150

average probability of undetected error is For very low weights, the problem of determining the
qk 1 corresponding term in the weight enumertor is tractable - we
pav (P) (1n
- (1 p)f) can fully characterize all codewords of a given low weight
- 1and thereby count them. The results for weight-2 and weight-
For systematic codes, a similar result is known [11], [8]: 3 codewords are summarized in the following propositions.
psYS(p) q (n-k)(1 (1- k) Proposition 1 The number of weight-2 codewords in an
OERS code is
Note that in either case, the average performance of a ran- -() (q 1)2 + ab(q-1) if d > 2
domly chosen code satisfies the conditions for both goodness
2, d 2
n
A/ = ') (q- 1) ifd= ,
and properness. 2 2
In this paper, we consider Over-Extended Reed-Solomon (2) (q 1) if d 1
(OERS) codes. From a practical point of view, these codes where a and b are integers such that n' = an + b, 0 < b < n.
are constructed by reusing a (shift register type) RS encoder
but allowing a longer input. Let C be a RS code over Fq with Corollary 2 If d > 2 and n. n', then A= (n) (q - 1)2. g
length n. q -C1 and minimum distance d. Then C can be
described as the set of polynomials c(e)such that
For example, if the OERS code has twice the length of the
original RS code, then A' = (q - 1)2 if d > 2.
c(z) = -r(x) + x d-1
(x) (') -r(x)+ u()(2)
(2)
Proposition 3 The number of weight-3 codewords in an
where u(x) is the data polynomial of degree at most n - d, OERS code is
and r(x) is the remainder of xd-lu(x) divided by g(x), the ((3) (q - 1) + (a)b) (q - 1)(q - 2) if d > 3
generator polynomial of C. An OERS code C' can then be J { ( -
(n' -2) ((a) (q -1) + ab)
defined simply by allowing u(x) in (2) to have degree higher A/ + q ((a) (q - 1) + (a)b) }(q - 1) if d = 3
than n - d, such that the length of the code is extended to | 1 2 i d-2
n' > n. This results in a linear (n' , n' -d + l) code over ]Fq,
q q
The rest of the paper is arranged as follows. In Section II, ( 3
- f d 2
we derive upper and lower bounds on the weight distribution of where a and b are integers such that n' = an + b, 0 < b < mm n.
OERS codes. In Section III, we apply the results of Section II
to obtain tooban bounds ond onn the hepobbliyofuneece.err.o probability of undetected error forprahtunesndgteetiewgtdsrbton The study of these special cases motivates a general ap-
OERS codes on q-ary symmetric channels. We then prove that proach to understanding the entire weight distribution of
f
.
tERS bondes onqare tight.mInmSectrio IV,hann
Wex pen iven
t OERS codes. The following two lemmas, though simple, are
these bounds are tight. In Section IV, an example iS givenl th'ai o uho h dsusohtflos
to demonstrate the effectiveness of the bounds. Section V
concludes the paper. Lemma 4 If c(x) C Fq[x] and deg(c) < n', then c(x) C C' if
II.
WEIGHT DISTRIBUTION
f
and only if Rxrl_1[c(x)] C C. :
First, a few remarks on notation. Throughout the rest of Lemma 5 For all c(x) Fq[3],
the paper, unless otherwise stated, C is a RS code over Fq wt(c(x)) > wt(RXn_ [c(X)]).
with length n = q - 1, minimum distance d, and generator
polynomial g(x); C' is the OERS code constructed from C Let Az denote the set of weight-i codewords of C' We are
. , . . - . . , ~~~~~~~interested in finding A =A'a for all 1. From Lemma 4,
with length n' > n. In most of our discussions, C and C' t
will be interpreted as subsets of Fq[X], the set of polynomials since 0 e C, if we define 13 = {c(x) C Fq[3] t(C(X))
with coefficients in Fq. If c(x) C Fq[X], then deg(c(x)) is ,deg(c(x)) n.. Therefore, all OERS codes contain codewords of tP~(i(n,r) := {6 C tPi: 16

The formula for B' can now be stated, as follows.
Proposition 7 For all i, 35
(3 3 .5
11
where '{l.j} =1 if 1 < j, 0 otherwise, a [= 2 ,bnj/, b-
Rn[~n'] and 1. = ,3
It can be shown thlat A4 13$ for i d.
Fig. 1. Breakdown of Pud (p) by weights for a (15, 13) RS code over F16.
Proposition 9 This code is good.
AX J (n)q id±l±+B/ if d. Fr,/n2(d-2). 4
(4) 3.5 Pud(P)
wrpoitha iojn. j b
i- l(t')(q-d)(q- i)i d [r'/r]d (5f
>5\i1 1), t
where l

can add up the corresponding fi (p)'s to get a lower bound. where
In particular, consider contribution from weight-2 codewords
alone. We already know how many weight-2 codewords there p Ed
are, so the following result is immediate. _ud (P) SBi q 1( P)
Proposition 11 For d > 2, a simple lower bound on Pud(p) ( [](d- 1)
IS Pv,d(p) > E
p1ud(p), + (1)d -
(i> i- P)n-i)
where
1n'] (d-1)
Puj)(p) a(() + b p2(1 p)n'-2 + ( )d K5 p (1-p)
and a and b are integers such that n' = a(q - 1) + b, 0 < b <
q -1. F- nnti] (d-2)i
Note that from (1), it is clear that as p -* 0, Pud(p) is dom- Pud(p) = B (q 1p
inated by the contribution from mininum weight codewords. / d-1
Therefore, the bound ud (p) is asymptotically tight as p -*0. + 1 1 i(1 -)n'i
In fact, we shall see later that, under some mild conditions, (q - l)d1 Vi jP P11
this bound is also tight for bounding the worst-case Pud.
nThe dP tohow that mOES codes
codes are not good. The message here is that OERS codes
We now use the bound p)(2) to show that most OERS where B' is given by (3), and Ki' by (6).
Corollary 14 For OERS codes,
the worst-case probability of
tend to have a disproportionately large number of weight-2 undetected error, max maxo 1 be fixed. If d > 3, then OERS
codes are not good when q is sufficiently large. 2 Proposition 16 For any given n'/n and d> 3, as q oo,
-max
The approximate analysis above also shows us that the p(2) prmax pnmax pd
degree of performance degradation from the good-code bound,
i.e. Cq-2 versus q- (d1), is most sensitive to d. The crucial Finally, we should note that since RS codes are usually
matter here is that the number of weight-2 codewords in an used for relatively short block lengths, the asymptotic analysis
OERS code does not depend on the number of parity check may seem less useful. However, it is clear that as long as
symbols (in this case d - 1). While Padg is expected to n' >[>[ n(d -1), i.e. n >> d -1, the asympotic behavior
decrease exponentially with d, P(2) is not affected. The gap is well approximated and is instructive for predicting how tight
in performance can be very pronounced even when d is only these bounds are. Since for storage systems, high rate codes
moderately larger than 3. are common, this condition is in many cases valid.
We now give upper and lower bounds on Pud (p) for OERS The asymptotic results can also be used to reduce complexcodes,
based on bounds we have obtained on their weight ity of the bounds. In particular, in the lower bound (5) onA'
distribution. it can be seen that B' is negligibly small and can be safely
Thus, in the expression of ud (p), we only need to
. . ~~~~~~~~~~~~ignored.
Proposition 13 For QERS codes, calculate B"'s up to i =d -1. (We could do the same for the
EPd(P) . Pud(p) . Pud(p), upper bound but it would make it an approximate bound.)
153

~~~~~~~~~~~~~~REFERENCES
TABLE I
10 ......................................BOUNDS ON pmdax FOR (2m+1 2, 2m+1 5) OERS CODES OVER 1F2
L Lower:bound--: b_un X -:- -:-: K......... :: -: ... ......... ... :--: nmax pmax -max (2)
ud ud ud ax
m 3 3.5 120e-03 3.3622e-03 3.7427e-03 3.2095e-03
- -- m 5 1.5004e-04 1.4946e-04 1.5054e-04 1.4550e-04
10
,.. .. ._ .. .. ..
m-6 3.24e0 3X10-5 35325 344e0
8.5262e-06 8.5241e-06 8.5279e-06 8.4573e-06
-6 - -----------------------------
-m8- 2.0979e-06 2.0981e-06 2.0895e-06
10 ._.
1 J. K. . . B...... .,k , , " l of
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theiry
in Table pertforsmance I. As an ihnnterm example erofproabilit how tight ofthese undetpected bounds
sner'ror.'.
are,
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between 2.0979 x 10-6 and 2.0981 x 10-6.
V. CONCLUSION
OERS codes are of practical interest as they have been
used in data storage systems. In this work we have examined
their performance in terms of probability of undetected error