By using coding and combinatorial techniques, an approximate formula for the weight distribution of decodable words of most linear block codes is evaluated. This formula is then used to give an approximate expression for the decoder error probability P(sub E)(u) of linear block codes, given that an error pattern of weight u has occurred. It is shown that P(sub E)(u) approaches the constant Q as u gets large, where Q is the probability that a completely random error pattern will cause decoder error

Cheung, K.-M.

English

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TDA_Progress Report 42-98
N90-12793
August 15, 1989
On the Decoder Error Probability of Linear Codes
K.-M. Cheung
Communications Systems Research Section
In this article, by using coding and combinatorial techniques, an approximate
formula for the weight distribution of decodable words of most linear block codes
is evaluated. This formula is then used to give an approximate expression for the
decoder error probability PE(u) of linear block codes, given that an error pattern
of weight u has occurred. It is shown that PE(u) approaches the constant Q as u
gets large, where Q is the probability that a completely random error pattern will
cause decoder error.
I. Introduction
Coding is used in a digital communication system to
detect and correct errors introduced in the data stream by
channel noise. An important parameter to evaluate the
performance of a code is its decoder error probability. Let
C be a linear (n,k,d) code over GF(q), and C -L be its
(n, n- k, d J-) dual code. Let t be the number of errors the
code is designed to correct. Let G be the generator matrix
of C. Let Au denote the number of codewords of weight u,
and Du denote the number of decodable words of weight
u. Decodable words are defined as all words lying within
distance t from a codeword. If the decoder is assumed to be
a bounded distance decoder, then the weight distribution
for the decodable words can be used to find the decoder
error probability of the code.
When a codeword _c E C is transmitted over a com-
munication channel, channel noise may corrupt the trans-
mitted signals. As a result, the receiver gets a corrupted
version of the transmitted codeword _c+ e_, where e_ is an
error pattern of some weight u. If u _ t, then a bounded
distance decoder on the receiver's end detects and corrects
the error e_, and recovers _c. If u > t, the decoder fails, and
it either
(1) Detects the presence of the error pattern e.c, but is
unable to correct it, or
(2) Misinterprets the received pattern c+e_ for some other
codeword c/if the received pattern falls into the radius
t of the Hamming sphere around d.
Case (2) is, in most cases, more serious than case (1).
This can occur when an error pattern e is of weight u >_
d- t. As pointed out in [1] and [2], if all error patterns of
weight u are equally probable, the decoder error probabil-
104
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ity given that an error pattern of weight u occurs, denoted
by PE(u), is given by the following expression:
Du
PE(U)-- (:)(q_ i) _ d-t < u < n (i)
In this article, by using combinatorial and coding tech-
niques, an approximate formula for the weight distribution
of decodable words for most linear block codes is evaluated.
This formula together with Eq. (1) gives an approximate
formula for the decoder error probability Pe(u) for most
linear block codes. It is also shown that
Re(u) 0
where Q is the probability that a completely random error
pattern will cause decoder error. That is,
Q _ (qk _ 1)V,(t) _ q_rV,,(t ) (2)
q,_
where r = n - k is the code's redundancy and Vn(t) =
t
_'_,=o (?)(q - 1) I is the volume of a Hamming sphere of
radius t.
II. Mathematical Preliminaries
In this section, combinatorial and coding techniques
required to derive the results in later sections are intro-
duced. These techniques are similar to those used in [6] to
obtain the weight distribution of linear block codes.
A. Principle of Inclusion and Exclusion
Let X be a set of N objects, and P(1), P(2),..., P(u)
be a set of u properties. Let N(il, i2, ..., it) be the number
of objects with properties P(il), P(i2),...,P(ir). The
number of objects N(0) with none of the properties is given
by [3]:
N(O) = N - _-_N(i) + _ N(il,i_) +... + (-1) r
i ix<i2
x y_ N(il,i2,...,i_)+...
Q<iz...<ir
+ (-1)UN(1,2,3,...,u) (3)
There are u + 1 terms on the right-hand side of Eq. (3),
with the 0th term representing the total number of objects
in X. If all terms beyond the rth term on the right-hand
side of Eq. (3) are ignored, then the resulting truncated
sum is an upper bound when r is even, or a lower bound if
r is odd. Thus the maximum error magnitude introduced
in the inclusion and exclusion formula by ignoring all terms
beyond the rth term does not exceed the magnitude of the
(r+ 1)th term. This fact will be used later to upper bound
the magnitude of the errors of the approximate weight dis-
tribution formula.
B. Facts on Coding Theory
A linear (., d) codeoverCF(q) canbe generated
by a k x n generator matrix G, not necessarily unique and
such that rank(G) = k. Let l be the maximum number
such that no I or fewer columns of G add to zero. Then
I < k (4)
Equality in Eq. (4) is achieved in the case of maximum
distance separable (MDS) codes. Since G is the parity-
check matrix of C "L, l = d "L- 1. Let colit, COil2,..., colii
be any j particular columns of G, j < i < k. It is obvious
that there exists a k x n generator matrix G' of C and a
k x k nonsingular matrix K such that
G' = KG (5)
and COil1, coli2,...,coli _ of G' form a k x j submatrix
of the form (.Io.) . This fact guarantees that given any
pattern ofj symbols on the il th, i2th,..., ij th coordinates,
the number of codewords with the j-symbol pattern on
the ilth, i2th, ..., i./th coordinates equals q_-J for j < I.
This fact is important in the next section to evaluate the
cardinalities of some sets of decodable words.
III. Derivation of Formulae
Let D be the set of decodable words of C. Let _dbe a
decodable word with Hamming weight u, u > n-l. Let the
coordinates ofd be indexed by {0, 1,..., n- 1}. Then d has
v zeros (v < l), where v = n - u. Let V be a set of v coor-
dinates, IYl = v. Let {il,i2,...,ij} C_ {0,1,...,n-1}-Y
be a set of j coordinates. Define S(il,i2,...,ij) = {d :
105
_d E D and d has zeros in VU {il, i2,...,ij}}. A decod-
able word _d E S(il,i2,...,ij) always has at least v + j
zeros. For 0 < j < l-v, the number of zeros in the de-
codable words of S(il,i2 .... ,i j) is less than or equal to
I. Now, since all words lying within the Hamming spheres
(with volume V,(t)) that surround codewords are decod-
able words, there are V,(t) disjoint cosets that contain de-
codable words. Each coset can be constructed by adding a
coset leader a (Hamming weight ofa < t) to each codeword
in C. Thus from the discussion in Section II.B, for each
of the V,(t) different coset leaders (each corresponding to
a coset), there are qk-v-j codewords in C which, when
added to the coset lender, give decodable words with zeros
in the v + j coordinates. The number of decodable words
in S(it, i2,..., ij) is then given by
IS(ix, i2,..., i_)l = qk-_-_ V,(t)
for o<_j<__-v (8)
For l-v+l < j < n-v-d+t, the number of zeros in
the decodable words of S(il,i2,...,ij) exceeds l, and ap-
parently there is no simple expression for 18(i1, i2 .... , i¢)l.
For n-v-d+t+ 1 < j < n-v-t, the number
of zeros in a decodable word is greater than or equal to
n - d + t + 1, but less than or equal to n - t. Thus any
decodable word in S(ix, i2,... ,ij) has weight less than or
equal to d - t - 1. It is not hard to see that the elements
of S(il,i2,...,ij) cannot be decoded into a codeword of
weight other than 0. Therefore, S(il,i2,...,ij) contains
all words of weight less than or equal to t in the coordi-
nates {0, 1, ...n - 1} - (V O {il,i2,... ,ij}). Thus,
IS(h, i2,..., ij)[ = qU-_
forn-v-t+l<j<n-v (8)
In the cases for 0 < j < l-v, n-v-d+t+l <
j < n-v-t, and n-v-t+l < j < n-v, the set
ii,i_,...,ij can be chosen arbitrarily from a set of u =
n - v coordinates. Thus for every choice of j, there are
(_.) sets S(ii, i2,..., ij). By the principle of inclusion and
exclusion, the number of decodable words with exactly v
zeros in V, which is denoted by D_, is:
D'v = IS(O)I+ (-1) E IS(ii)l +... + (-1) r
it
x y_ IS(ii,i2,... ,ir)l
ia<i_<...<i,
+... + (-1)"-_s(ii,i2,...,i,__)
I-v n-v-d+t
= E(-1) "/(_)q'-_-JV_(t)+ E (-1)j
j=0 j=l-v+l
× E IS(ii,i2, ... ,ij)l
il<i2<...<ij
.. (;)±()+ E (-1)J n-v-j (q_l)ii
jmn-v-dTt4-i i=0
IS(il,i2,...,ij)l=_-_(u_j)(q-1) i
i=O
"+ (_1)i qU- (91
jmn-v-t+l
forn-v-d+t+l<j<n-v-t (7)
For n-v-t+1 < j < n-v, since j is greater than or equal
to n - v - t + 1, the number of zeros v + j is greater than
or equal to n - t + 1. Therefore, the number of nonzero
components is less than t. Thus, all words with zeros on
V U {il, i2,..., ij} are decodable and thus
If all the terms beyond the l-v-1 terms are ignored in the
above inclusion and exclusion formu.la, Eq. (9) is reduced
to
I--v--1
lYV= E(-1)J(_)q'-_-JV,,(t)-bE1 (10)
j=0
106
where for u > max{n-l, _ql(n - l) - 1}.
El - (-1)Z-V(l U_v)qk-tVn(t )
+
j=l--v+l il<...<ij
and IEll _< (t:v)qk-tVn(t) (from the discussion in Section
II.A). If E2 = _-_;=t_v(-1)J(_.)qk-_-JVn(t)is added and
subtracted from Eq. (10), one has
(q - 1)u Vn(t) + E1 -t- E2 (11)
n_- q_-_-_
If (,_v)q -> (,-_+1), that is, if u > q_:_A(n - l) - 1, E2 is a
sum of terms with alternate signs and descending magni-
_ ( _ _ k-tVn(t). Thustude. Then [E21 < _t- )q
D_, - (q - 1)" Vn(t) + E (i2)
qn-k
where E = E1 + E2 and IEI < 2(t___)qk-_v.(t). D'v can
thus be approximated by _Vn(t), and the goodness
of approximation depends on how small the ratio R =
Eli(q- 1)Uq-(n-k)Vn(t)] is. By using the upper bound on
[El, an upper bound on this ratio is given by
2(n_ t)q k-t
R< (q_ 1)" (13)
Strictly speaking, the derivation of Eq. (14) is valid
only for u > max{n - l, q_:9-(n - l) - 1}. However, it is
observed that in most cases Eq. (14) is also a close ap-
proximation to Du for u considerably smaller than n - l
(as in the case of Reed-Solomon codes). The upper bound
of R derived above has a denominator term (q - 1) _' and
this indicates that this approximation formula is good for
nonbinary linear codes, and is not useful for binary linear
codes. The looseness of this approximation for binary lin-
ear codes is best illustrated by extended binary codes that
have only even weights. In the case of binary primitive
codes, Kasami et al. [4] generalized Sidel'nikov's approach
[5] and showed that the weights of most binary primitive
codes have approximate binomial distribution.
Cheung [6] later showed this is also true for nonbinary
codes. It is conjectured in this article that the approximate
Eq. (14) for the weight distribution of decodable words is
also good for binary primitive codes. For nonbinaxy linear
codes, the upper bound on R shows that the approxima-
tion in Eq. (14) is particularly good for codes with large
alphabet sets. The upper bound on R for the (31,15,17)
Reed-Solomon code over GF(32) is given in Table 1. The
weight distribution of decodable words and its approxima-
tion (using Eq. 14) of the (31,15,17) Reed-Solomon code
are given in Table 2.
Given the approximate formula of Du, an approxi-
mate decoder error probability PE(U) is obtained by sub-
stituting Eq. (14) into Eq. (1). It is observed that PE(u)
approximates the constant Q = q-rVn(t) as u gets large,
where Q is the probability that a completely random error
pattern will cause decoder error. An upper bound of R
given by Eq. (13) shows that PE(u) approaches Q "nearly
exponentially" (for nonbinary codes) as u increases.
Since v _< l, there are (_) = (:) ways to choose v zeros
from {0,1,... ,n- 1}. Then Du can be approximated by
the following expression:
D_ = Z DIv_q-('-k)(:) (q-1)_V"(t) (14)
IVl=n-u
IV. Conclusion
In this article, by using the inclusion and exclusion
principle, an approximate formula for the weight distri-
bution of decodable words of most linear block codes is
derived. The decoder error probability PE(u), which is a
function of D,, is then shown to approach the constant
Q as u gets large, where Q is the probability that a com-
pletely random error pattern will cause decoder error.
107
References
[1] R. J. McEliece and L. Swanson, "On the Decoder Error Probability for Reed-
Solomon Codes," IEEE Trans. Inform. Theory, vol. IT-32, pp. 701-703, 1986.
[2] K. Cheung, "More on the Decoder Error Probability of Reed-Solomon Codes,"
to appear in IEEE Trans. Inform. Theory.
[3] R. Stanley, Enumerative Combinatorics, Vol. I, Monterey, California:
Wadsworth and Brooks-Cole, 1986.
[4] T. Kasami, T. Pujiwara, and S. Lin, "An Approximation to the Weight Distri-
bution of Binary Linear Codes," IEEE Trans. lnform. Theory, vol. IT-31, no.
6, pp. 769-780, November 1985.
[5] V. M. Sidel'nikov, "Weight Spectrum of Binary Bose-Chaudhuri-Hocquenghem
Codes," Probl. Peredachi Inform., vol. 7, no. 1, pp. 14-22, January-March,
1971.
[6] K. Cheung, "The Weight Distribution and Randomness of Linear Codes," TDA
Progress Report 4g-97, vol. January-March 1989, Jet Propulsion Laboratory,
Pasadena, California, pp. 208-215, May 15, 1989.
108


On the decoder error probability of linear codes

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