Chapter 1 Introduction
2.2 Average Binary Image of Reed-Solomon Codes
The binary image Cb of an (n, k) code C over F2m is obtained by representing each symbol by an m-dimensional binary vector in terms of a basis of the field [75]. The weight enumerator of Cb will vary according to the basis used. In general, it is also hard to know the weight enumerator of the binary image of a certain Reed-Solomon code obtained by a specific basis representation (e.g., [67, 15]). For performance analysis, one could average the performance over all possible binary representations of C. By assuming that the all such representations are equally probable, it follows that the distribution of the bits in a nonzero symbol follows a binomial distribution
and the probability of having iones in a nonzero symbol is 2m1−1
¡m
i
¢.The generating function of the average weight enumerator of the binary image of a nonzero symbol is
F(Z) = Xm
i=1
1 2m−1
µm i
¶
Zi = (1 +Z)m−1
2m−1 , (2.6)
where the power of x denotes the binary weight and the all zero vector is excluded since the binary weight of a nonzero symbol is at least one. Suppose a codeword has w nonzero symbols, and the distribution of the ones and zeros in each symbol is independent from other symbols, then the possible binary weight,b, of this codeword ranges fromwtomw. Since there areE(w) codewords with symbol Hamming weight w, then the average binary weight generating function can be derived by
E˜Cb(X) = Xnm
b=0
E(b)X˜ b (2.7)
= EC(X)¯
¯X:=F(X) (2.8)
= Xn
h=0
E(h)
(2m−1)h ((1 +X)m−1)h. (2.9) A closed form formula for the average binary weight enumerator (BWE) is
E(b) = Coeff˜
³E˜Cb(X),Xb
´
(2.10)
= Xn
w=d
E(w) (2m−1)w
Xw
j=0
(−1)w−j µw
j
¶µjm b
¶
; b≥d. (2.11)
These results apply to any maximum distance separable code defined over Fq, where q = 2m and not necessarily an RS code. Widely used RS (MDS) codes have a code length n = 2m −1. In such a case the BWE derived in (2.10) agrees with the average BWE of a class of GRS codes [94]. In other words two ensembles have the same weight enumerator; the first ensemble is the ensemble of all possible binary
in the field. It is easy to see that Go = 1 and that ˜E(b) = 0 for 0< b < d.
By substituting forE(w), forb ≥d, the binary weight enumerator (BWE) is given by
E(b) = (q˜ −1) Xn
w=d
µ q q−1
¶wµ n w
¶
w−dX
v=0
(−1)v
µw−1 v
¶
Xw
j=db/me
(−1)w−j µw
j
¶µjm b
¶
q−(d+v)
. (2.12)
Although it is easy to evaluate the above formula, the term ¡jm
b
¢ may diverge numerically for large j. Using the Stirling approximation for ¡jm
b
¢ [74], ˜E(b) could be approximated as
E(b)˜ ≈ Xn
w=d
(q−1) µ q
q−1
¶wµ n w
¶w−dX
v=0
(−1)v
µw−1 v
¶ Xw
j=db/me
F(j), (2.13)
where
F(j) =
(−1)w−j¡w
j
¢2λ(j), j > b/m
(−1)w−j¡w
j
¢2−m(d+v), j =b/m
, (2.14)
and λ(j) =m(jH(ψb,j)−d−v)− 12log2(2πjmψb,j(1−ψb,j)) for ψb,j = b/jm and q = 2m. These bounds could be further simplified (and thus loosened) by observ- ing that for n ≤q−1,
1≤ µ q
q−1
¶w
≤ µ q
q−1
¶q−1
≤ lim
q→∞
µ q q−1
¶q−1
=e, (2.15)
and substituting in (2.13).
0 5 10 15 20 10−2
10−1 100 101 102 103 104
Binary Weight
Weight Enumerator
Ensemble Average Binary Weight Enumerator for the (7,5) RS Code
True Average
Approximate Average Normalized Binomial
Figure 2.1: True BWE versus the averaged BWE for the (7,5) RS code over F8.
is labeled “Approximate Average.” It is observed that a good approximation of the average binary weight enumerator for h ≥d is the normalized binomial distribution which corresponds to a random code with the same dimension over Fq
E(h)˜ ≈q−(n−k) µmn
h
¶
. (2.16)
This observation can be somehow justified by the central limit theorem, where the binary weight of a codeword is a random variable which is the sum ofn independent random variables corresponding to the binary weights of the symbols. For large n, the distribution of the binary weight is expected to converge to that of random codes.
The following theorem shows that the average BWE can be upper bounded by a
³ q q−1
´(n−k)
multiple of the above approximation.
Theorem 2.1. The average binary weight enumerator is upper bounded by
E(h)˜ ≤(q−1)−(n−k) µmn
h
¶ .
Proof. An upper bound on the symbol weight enumerator of an (n, k, d) MDS code defined over Fq is [79, (12)]
E(w)≤ µn
w
¶
(q−1)w−d+1; w≥d. (2.17)
Substituting in (2.10) it follows that for b≥d
E(b)˜ ≤(q−1)k−n Xn
w=d
µn w
¶
Xw
j=db/me
(−1)w−j µw
j
¶µjm b
¶
. (2.18)
By doing a change of variables α=mj and changing the order of summations
E(b)˜ ≤ (q−1)k−n Xn
w=d
Xmw
α=b
(−1)w−j µn
w
¶µ w α/m
¶µα b
¶
= (q−1)k−n Xnm
α=b
(−1)−mα µα
b
¶ Xn
w=max(mα,d)
(−1)w µn
w
¶µ w α/m
¶
≤ (q−1)k−n Xnm
α=b
(−1)−mα µα
b
¶ Xn
w=mα
(−1)w µn
w
¶µ w α/m
¶ .
From the identity¡n
m
¢¡m
p
¢=¡n
p
¢¡n−p
m−p
¢ it follows thatPn
k=m(−1)k¡n
k
¢¡k
m
¢ = (−1)mδnm where δn,m is the Kronecker delta function. It follows that
E(b)˜ ≤ (q−1)k−n Xnm
α=b
µα b
¶ δmα,n
= (q−1)k−n µmn
b
¶ ,
which completes the proof.
In Figure 2.2, we plot the ensemble average weight enumerator of (2.10) and compare it with the weight enumerator of a random code with the same dimension (2.16). We also compare it with the simple upper bound of Theorem 2.1. It is observed that the upper bound of Theorem 2.1 is fairly tight and that a good approximation for the ensemble weight enumerator is that of random codes. In fact, as length of the code (and the size of the finite field) tend to infinity
E(h)˜ ≤
µ q q−1
¶(n−k)
q−(n−k) µmn
h
¶
(2.19)
≤ e2−m(n−k) µmn
h
¶
(2.20)
≤ e
p2πmnλ(1−λ)2mn(H2(λ)−1+R), (2.21)
where b =λmn, R =k/n is the code rate and H2(λ) is the binary entropy function.
0 0.2 0.4 0.6 0.8 1 10
−3010
−2010
−1010
010
1010
2010
30Relative Binary Weight
Weight Enumerator
Ensemble Average Binary Weight Enumerator of the (31,15) RS Code
Average
Normalized Binomial Upper Bound
Figure 2.2: The ensemble weight enumerator of the (31,15) RS code over F32. The ensemble average weight enumerator of (2.10) is compared with the weight enu- merator of the random code (2.16) and the upper bound of Theorem 2.1. They are labeled “Average,” “Normalized Binomial” and “Upper Bound” respectively.
The last inequality follows from the Stirling’s inequality [74, p. 309]. Let the asymp- totic weight enumerator exponent of a code C, of length N and weight enumerator EC, be defined as
Ξ(λ)= lim∆
N→∞
log2(EC(λN))
N . (2.22)
It follows that the asymptotic weight enumerator exponent of the ensemble of binary images of Reed-Solomon codes is
Ξ(λ) =˜ n→∞lim
m→∞
log2
³E(λmn)˜
´
mn
≤ n→∞lim
m→∞
log2(e)− 12log2(mn)− 12log2(2πλ(1−λ))
mn +H2(λ)−1 +R
= H2(λ)−(1−R). (2.23)
In other words, as the code length and the finite field size tend to infinity, the weight enumerator of the ensemble of binary images of an RS code approaches that of a random code.
The error-correcting capability of a code relies a lot on the minimum distance of the code, which will be analyzed in the next section.