ar
X
iv
:0
80
1.
39
26
v1
[
cs
.IT
]
25
Ja
n 2
00
8
On the Weight Distribution of the
Extended Quadratic Residue Code of
Prime 137
C. Tjhai, M. Tomlinson, M. Ambroze and M. Ahmed
Fixed and Mobile Communications Research
University of Plymouth
Plymouth, PL4 8AA, United Kingdom
Post-print of 7th International ITG Conference on Source and
Channel Coding, Ulm, 14–16 January 2008
Abstract
The Hamming weight enumerator function of the formally self-dual
even, binary extended quadratic residue code of prime p = 8m + 1 is
given by Gleason’s theorem for singly-even code. Using this theorem,
the Hamming weight distribution of the extended quadratic residue
is completely determined once the number of codewords of Hamming
weight j Aj , for 0 ≤ j ≤ 2m, are known. The smallest prime for
which the Hamming weight distribution of the corresponding extended
quadratic residue code is unknown is 137. It is shown in this paper that,
for p = 137 A2m = A34 may be obtained without the need of exhaustive
codeword enumeration. After the remainder of Aj required by Gleason’s
theorem are computed and independently verified using their congru-
ences, the Hamming weight distributions of the binary augmented and
extended quadratic residue codes of prime 137 are derived.
1 Introduction
The Hamming weight distribution of a linear error correcting code is of
practical and theoretical interest. It provides a great deal of information on
the code capability in detecting errors and in correcting errors or erasures.
The complexity of computing the Hamming weight distribution of a code
is exponential. In general, the computation requires one to enumerate all
codewords of the code; or to enumerate all codewords of the dual and apply
the MacWilliams identity.
Since the birth of coding theory, various algebraic error correcting
codes have been discovered. One classic family of such codes is the family
of quadratic residue (QR) codes, which has rich mathematical structure and
good error correcting capability. Despite having these advantages, the con-
struction of its algebraic decoder is non trivial. Due to the existence of rich
mathematical structure, there are considerable restrictions on the weight
1

structure of this family of codes and therefore it is not necessary to enumer-
ate all codewords or those of the dual in computing the Hamming weight
distribution. In fact, by knowing a fraction of the Hamming weight distri-
bution, the complete distribution can be obtained. Recently, this method
has been used by Gaborit et al [1] to obtain the Hamming weight distribu-
tions of binary extended QR codes of primes 73, 89, 97, 113 and 1271. In
our previous work [2, 3], we have evaluated the Hamming weight distribu-
tions of the extended QR codes of primes 151 and 167. The smallest prime
for which the Hamming weight distribution of the corresponding extended
QR code is not known in 137 and in this paper, its Hamming weight distri-
bution is evaluated. We show that even smaller fraction of the Hamming
weight distribution is sufficient to derive the complete Hamming weight
distribution.
The remainder of this paper is organised as follows. Section 2 gives
the definition and notation that we use in this paper–including a brief recall
of the binary QR codes. Section 3 discusses the modular congruence of the
number of codewords of a given Hamming weight and the Hamming weight
distribution of the extended QR code of prime 137 is derived in Section 4.
2 Definition and Notation
Let Fn2 be a space of vector of length n whose elements take value over
F2 (binary field). An [n, k, d] binary linear code C of length n, dimension
k and minimum Hamming distance d, is a k-dimensional subspace of Fn2 .
Let x,y ∈ Fn2 , the scalar product of these two vectors is defined as x · y =
∑n−1
j=0 xjyj (mod 2). Given a code C, the dual code is defined as C⊥ = {c⊥ |
c · c⊥ = 0 for all c ∈ C and c⊥ ∈ Fn2}. The hull of a code C is defined as
H (C) = C ∩ C⊥.
The Hamming weight of a vector v ∈ Fn2 , denoted by wtH(v), is the
number of its non zero coordinates and the minimum Hamming distance of
C is simply the smallest Hamming weight of all codewords in C. Throughout
this paper, we deal exclusively with Hamming space and for convenience,
the word “Hamming” shall be omitted. The weight enumerator function of
C is given by
AC(z) =
n
∑
j=0
Ajzj (1)
where z is an indeterminate and Aj is the number of codewords of weight
j. The distribution of Aj for 0 ≤ j ≤ n is called the weight distribution of a
code.
Given a vector v ∈ Fn2 of even weight, if wtH(v) ≡ 0 (mod 4), it is
termed doubly-even; otherwise wtH(v) ≡ 2 (mod 4) and it is termed singly-
even. An even code is one which has codewords of even weight only. A
code C is called self-dual if C = C⊥. A self-dual code may be doubly-even
if the weight of all codewords is divisible by 4 or singly-even if there are
1The Hamming weight distribution of that of prime 151 is also given in [1], but we have
shown that this result has been incorrectly reported, refer to [2] for the detailed discussion.
2

codewords whose weight is congruent to 2 (mod 4). In addition to self-dual
code, there also exists formally self-dual code. A code C is termed formally
self-dual if C 6= C⊥ but AC(z) = AC⊥(z).
2.1 Quadratic Residue Codes
In this subsection, a brief summary of QR codes over F2 is given [4]. Binary
QR codes are cyclic codes of prime length p where p ≡ ±1 (mod 8). Let
Q and N be sets of quadratic residue and non quadratic residue modulo p
respectively. Given a prime p, there are four QR codes denoted by Qp, Np,
Qp and N p. If α is a primitive p-root of unity, the generator polynomial
of the [p, (p + 1)/2, d − 1] augmented QR codes Qp and Np contains roots
whose exponents are element of Q and N respectively. The [p, (p − 1)/2, d]
expurgated QR codes Qp and N p contain, in their generator polynomial, α0
in addition to the roots of the respective augmented QR codes. Note that
Qp (resp. Qp) is permutation equivalent to Np (resp. N p).
If p ≡ −1 (mod 8), Q⊥p = Qp and as such the [p + 1, (p + 1)/2, d]
extended QR code Qˆp is self-dual and doubly-even. For p ≡ 1 (mod 8),
Q⊥p = N p and therefore Qˆp 6= Qˆ⊥p but AQˆp(z) = AQˆ⊥p (z) implying the corre-
sponding extended QR code is formally self-dual.
In this paper, we are interested in the QR codes where p ≡ 1 (mod 8),
in particular p = 137. Since the extended code is formally self-dual, the re-
strictions on the weight structure imposed by Gleason’s theorem for singly-
even code applies. This implies that for a given prime p = 8m+1, the weight
enumerator function AQˆp(z) is given by [5]
AQˆp(z) =
m
∑
j=0
Kj(1 + z2)4m−4j+1{z2(1 − z2)2}j (2)
for some integer Kj . Equation (2) shows that the complete weight distribu-
tion can be derived once the first m even terms of Aj (A0 = 1 by definition)
are known. Note that Qˆp is an even code and thus Aj = 0 for odd integer j.
3 Congruence of the Number of Codewords of
a Given Weight
It is known in the literature that the automorphism group of Qˆp, denoted by
Aut(Qˆp), contains the projective special linear group PSL2(p) [4]. This lin-
ear group is generated by a set of permutations on the coordinates (∞, 0, 1, . . . , p−
1) of the form y → (ay + b)/(cy + d) where a, b, c, d ∈ Fp, y ∈ Fp ∪ {∞} and
ad − bc = 1. This set of permutations may be produced by the transforma-
tions2 S : y → y + 1 and T : y → −y−1. The knowledge of the automorphism
group of a code may be exploited to characterise the weight distribution of
the code.
2In some cases, we can see that, in addition to S and T , the transformation V : y → ρ2y
where ρ is a generator of Fp also generates the desired permutation of PSL2(p). However,
strictly speaking, V is redundant since V = TSρTSµTSρ where µ = ρ−1 (mod p).
3

Let Aut(Qˆp) ⊇ PSL2(p) = H, the number of weight j codewords Aj
can be categorised into two classes: one which contains all weight j code-
words that are invariant under some element of H and another which con-
tains the rest. Given a codeword of Qˆp that is not invariant under some
element of H, applying all |H| = 12p(p2 − 1) permutations will result in |H|
distinct codewords of Qˆp. In other words, the latter class forms orbits of size
equal to the cardinality of PSL2(p). Let Aj(H) denote the number of weight
j codewords which are invariant under some element of H, we may write
Aj = nj · |H| + Aj(H)
≡ Aj(H) (mod
1
2
p(p2 − 1))
(3)
for nj ∈ Z∗ = {0} ∪ Z+ i.e. non negative integer. Since |H| can be factorised
as H = ∏i qeii where qi is a prime and ei is a positive integer, it is shown in
[6] that Aj(H) may be obtained by applying the Chinese Remainder The-
orem to Aj(Sqi) (mod q
ei
i ) for all primes qi that divide |H|. Note that Sqi
is the Sylow-qi-subgroup of H and Aj(Sqi ) is the number of codewords of
weight j fixed by some element of Sqi .
For each prime qi, in order to compute Aj(Sqi), the subcode which is
invariant under some element of Sqi needs to be obtained. For odd primes
qi, Sqi is cyclic and there exists
[
a b
c d
]
∈ H, for some integers a, b, c, d, which
generates cyclic permutation of order qi. Thus, it is straightforward to ob-
tain the invariant subcode and the corresponding Aj(Sqi). On the other
hand, if qi = 2, S2 is a dihedral group of order 2s, where s is the highest
power of 2 that divides |H|, and Aj(S2) is given by [6]
Aj(S2) ≡ (2s−1 + 1)Aj(H2) − 2s−2Aj(G04) − 2s−2Aj(G14) (mod 2s), (4)
where H2 and Gi4, for i = 0, 1, are subgroups of order 2 and 4 respectively,
which are contained in S2. Let P ∈ H of order 2s−1 and T =
[
0 −1
1 0
]
∈ H of
order 2, it is shown in [6] that H2 = {1, P 2
s−2} and the non cyclic subgroup
Gi4 = {1, P 2
s−2
, P iT, P 2
s−2+iT }.
4 The Weight Distribution
Following Gleason’s theorem, see (2), the weight distribution of the binary
extended QR code of prime 137 is given by
AQˆ137(z) =
17
∑
j=0
Kj(1 + z2)69−4j(z2 − 2z4 + z6)j . (5)
Since A0 = 1 and the minimum distance of Qˆ137 is 22, only A2j , for 11 ≤ j ≤
17, are required in order to deduce AQˆ137(z) completely. Note that each A2j
determines Kj for some integer j. However, following the idea in [6] which
has been relatively forgotten, K17 may be determined without the need of
exhaustively computing A34 as shown in this section.
Let us first deduce the modular congruence of A2j , for 11 ≤ j ≤ 17, of
Qˆ137. Some of these congruences have been given in the authors’ previous
4

work [2], but are restated in the following to make the paper self-contained.
For p = 137, it is clear that |H| = 23 ·3 ·17 ·23 ·137 = 1285608. Let P =
[
0 37
37 31
]
and let
[
0 1
136 1
]
,
[
0 1
136 6
]
and
[
0 1
136 11
]
be generators of permutation of orders 3,
17 and 23 respectively. It is not necessary to find a generator that generates
permutation of order 137 as it fixes the all zeros and all ones codewords
only. Subcodes that are invariant under H2, G04, G14, S3, S17 and S23 are
obtained and the number of weight 2j, for 11 ≤ j ≤ 17, codewords in these
subcodes are then computed. The results are tabulated as follows, where k
denotes the dimension of the corresponding subcode,
H2 G04 G
1
4 S3 S17 S23 S137
k 35 19 18 23 5 3 1
A22 170 6 6 0 0 0 0
A24 612 10 18 46 0 0 0
A26 1666 36 6 0 0 0 0
A28 8194 36 60 0 0 0 0
A30 34816 126 22 943 0 0 0
A32 114563 261 189 0 0 0 0
A34 343453 351 39 0 2 0 0
.
For p = 137, (4) becomes
A2j(S2) ≡ 5A2j(H2)− 2A2j(G04) − 2A2j(G14) (mod 8)
and using this formulation, the following congruences
A22(S2) = 2 (mod 8)
A24(S2) = 4 (mod 8)
A26(S2) = 6 (mod 8)
A28(S2) = 2 (mod 8)
A30(S2) = 0 (mod 8)
A32(S2) = 3 (mod 8)
A34(S2) = 5 (mod 8)
are obtained.
Combining all the above results using the Chinese-Remainder-Theorem,
it follows that
A22 = n22 · 1285608 + 321402
A24 = n24 · 1285608 + 1071340
A26 = n26 · 1285608 + 964206
A28 = n28 · 1285608 + 321402
A30 = n30 · 1285608 + 428536
A32 = n32 · 1285608 + 1124907
A34 = n34 · 1285608 + 1143813
(6)
for some non negative integers n2j .
Let G be the generator matrix of the half-rate code Qˆ137. In order to
efficiently count the number of codewords of weight 2j, two full-rank gener-
ator matrices, say G1 and G2, which have pairwise disjoint information sets
5

are required. These matrices can be easily obtained by performing Gaus-
sian elimination on G to produce G1 = [I|A] and repeating the process on
submatrix A to produce G2 = [B|I]. For each of these full-rank matrices,
we need to enumerate as many as
j
∑
i=0
(
69
i
)
codewords and count the number of those of weight 2j. The efficiency of
enumeration may be improved by employing the revolving door combina-
tion generator algorithm [7], which has the property that in two successive
combination patterns, there is only one element that is exchanged. In ad-
dition to this, the revolving door algorithm also has a nice property that
allows the enumeration to be realised on grid computer, see Appendix A.1.
We have evaluated A2j , for 11 ≤ j ≤ 16, using a grid of approximately 1500
computers and the results are given below
A22 = 321402
A24 = 2356948
A26 = 21533934
A28 = 490138050
A30 = 6648307504
A32 = 77865259035.
(7)
Comparing (6) and (7), it can be clearly seen that3 n22 = 0, n24 = 1, n26 = 16,
n28 = 381, n30 = 5171 and n32 = 60566. The non negative integer solutions
of n2j give an indication that the corresponding A2j has been accurately
computed.
We now show that A34 is known. It is worth noting that knowing
A34, based on the arguments on codeword counting given above, signifi-
cantly reduces the complexity of computing AQˆ137 (z). Consider Gleason’s
formulation given in (5), if we take its first derivative with respect to z, we
have
d
dz
AQˆ137 (z) =
17
∑
j=0
Kj(1 + z2)68−4j(z2 − 2z4 + z6)j−1
{
2(69− 4j)z(z2 − 2z4 + z6)+
j(1 + z2)(2z − 8z3 + 6z5)
}
(8)
3Note that A2j , for 11 ≤ j ≤ 16, have also been given in [1], however, A30 and A32 have
been incorrectly reported as demonstrated in [2].
6

which may be expanded as
d
dz
AQˆ137(z) = (1 + z
2)68K0+
(1 + z2)64
{
130z(z2 − 2z4 + z6)+
(1 + z2)(2z − 8z3 + 6z5)
}
K1+
(1 + z2)60(z2 − 2z4 + z6)
{
122z(z2 − 2z4 + z6)+
2(1 + z2)(2z − 8z3 + 6z5)
}
K2+
...
(z2 − 2z4 + z6)16
{
2z(z2 − 2z4 + z6)+
17(1 + z2)(2z − 8z3 + 6z5)
}
K17.
(9)
From (9), we can see that the terms that involve Kj for 0 ≤ j ≤ 16 become
zero if we set z = i =
√
−1. Thus,
d
dz
AQˆ137(z)



z=i
= 2i(i2 − 2i4 + i6)17K17
= −i235K17. (10)
Since Aut(Qˆp) is doubly-transitive, given A2j of an extended QR code
Qˆp, the number of codewords of weight 2j − 1 and 2j in the augmented
code Qp are 2jp+1A2j and
p+1−2j
p+1 A2j respectively. Following [8], the weight
enumerator function of Q137 may be written in terms of that of Qˆ137 as
follows
AQ137 (z) = AQˆ137(z) +
(
1 − z
138
)
d
dz
AQˆ137 (z). (11)
From (5), it is obvious that AQˆ137 (z)



z=i
= 0 and therefore (11) becomes
AQ137(z)



z=i
= −i1 − i
138
235K17. (12)
The expurgatedQR codeQ137 is an even code and following [4],Q⊥137 =
N137. We can see that the exponents of the zeros of Q137 are in the set
Q ∪ {0}, whereas those of N137 are in the set N , and thus the hull of Q137
has dimension zero. It follows from [9, Lemma 7.8.3 pp. 276] that the code
Q137 may be decomposed into an orthogonal sum of either 34 subcodes each
consisting of three doubly-even and one singly-even codewords; or 33 sub-
codes each consisting of three doubly-even and one singly-even codewords,
in addition to one subcode containing one doubly-even and three singly-
even codewords. As a consequence, if Ww denotes the number of codewords
of weight congruent to w (mod 4) in Q137, we have, see [9, Theorem 7.8.6
pp. 277]
W0 −W2 = ±234. (13)
7

Note that this result also holds for Q137 as Q137 is the even weight subcode
of Q137. Since all ones codeword 1p ∈ Q137, it follows that
W1 −W3 = ±234 (14)
for the augmented QR code. Substituting z with i in the weight enumerator
function of Q137, we have
AQ137 (z)



z=i
= A0 + iA1 −A2 − iA3+
A4 + iA5 −A6 − iA7+
...
−A130 − iA131 + A132 + iA133
−A134 − iA135 + A136 + iA137
=
[
∑
j≡0 mod 4
Aj −
∑
j≡2 mod 4
Aj
]
+
i
[
∑
j≡1 mod 4
Aj −
∑
j≡3 mod 4
Aj
]
= [W0 −W2] + i[W1 −W3]
and thus, following (13) and (14),
AQ137(z)



z=i
= ±234(1 + i). (15)
Equating (12) and (15),
−i1 − i
138
235K17 = ±234(1 + i),
we arrive at
K17 = ∓69. (16)
Using (7), A2j = 0 for 1 ≤ j ≤ 10 and A0 = 1, Kj for 0 ≤ j ≤ 16 are
determined. Substituting these into (5) and equating the coefficients of z34
with A34, we have
A34 = 771068968296+ K17. (17)
Consider the case for K17 = −69, A34 = 771068968227. Comparing this A34
with the congruence given in (6), it follows that n34 6∈ Z∗ and hence this
rules out the possibility of K17 = −69. If K17 = 69, however,
A34 = 771068968365 (18)
and it follows that n34 = 599769 ∈ Z∗, indicating that K17 is indeed 69.
Now we have determined A34 (and hence K17) without exhaustively
counting the number of codewords of weight 34 in Qˆ137. The weight dis-
tribution of Qˆ137 can be straightforwardly deduced from (5) and so is that
of Q137 from (11). The weight distributions of the augmented and also the
extended QR code of prime 137 are tabulated in Table 1. Note that since
the weight distributions are symmetrical, only the first half terms are tab-
ulated.
8

Acknowledgements
The authors wish to thank the PlymGRID team of the University of Ply-
mouth for providing the high performance computing resources.
References
[1] P. Gaborit, C.-S. Nedeloaia, and A. Wassermann, “On the weight enu-
merators of duadic and quadratic residue codes,” IEEE Trans. Inform.
Theory, vol. 51, pp. 402–407, Jan. 2005.
[2] C. Tjhai, M. Tomlinson, R. Horan, M. Ahmed, and M. Ambroze, “Some
results on the weight distributions of the binary double-circulant codes
based on primes,” in Proc. 10th IEEE International Conference on Com-
munications Systems, (Singapore), 30 Oct.–1 Nov 2006.
[3] C. Tjhai, M. Tomlinson, R. Horan, M. Ahmed, and M. Ambroze, “On
the efficient codewords counting algorithm and the weight distribu-
tion of the binary quadratic double-circulant codes,” in Proc. IEEE In-
formation Theory Workshop, (Chengdu, China), pp. 42–46, 22–26 Oct.
2006.
[4] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting
Codes. North-Holland, 1977.
[5] E. M. Rains and N. J. A. Sloane, “Self-Dual Codes,” in Handbook of
Coding Theory (V. S. Pless and W. C. Huffman, eds.), Elsevier, North
Holland, 1998.
[6] J. Mykkeltveit, C. Lam, and R. J. McEliece, “On the weight enu-
merators of quadratic residue codes,” JPL Technical Report 32-1526,
vol. XII, pp. 161–166, 1972.
[7] A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms for Computers
and Calculators. Academic Press, London, 2nd ed., 1978.
[8] J. H. van Lint, “Coding theory,” in Lecture Notes in Mathematics
No. 201, Springer, Berlin, 1970.
[9] W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting
Codes. Cambridge University Press, 2003. ISBN 0 521 78280 5.
[10] H. Lüneburg, “Gray codes,” Abh. Math. Sem. Hamburg, vol. 52,
pp. 208–227, 1982.
[11] D. E. Knuth, The Art of Computer Programming, Vol. 4: Fascicle 3:
Generating All Combinations and Partitions. Addison-Wesley, 3rd ed.,
2005. ISBN 0 201 85394 9.
9

A Appendix
A.1 Parallel Realisation of Codeword Enumeration
In this appendix, a method to enumerate codewords in parallel is described
and for a detailed description, refer to [7, 10, 11]. Let Cst denote the combi-
nation of t out of s elements with the combination pattern represented by
an ordered set atat−1 . . . a1, where a1 < a2 < . . . < at−1 < at. A pattern is
said to have rank r if this pattern appears as the (r + 1)th element in the
list of all Cst combinations. Here, it is assumed that the first element in the
list of all Cst combinations has rank 0. The combination Cst , which follows
the revolving door constraint and has an ordered set pattern, exhibits the
following property
Cst ⊃ Cs−1t ⊃ . . . ⊃ Ct+1t ⊃ Ctt .
Consequently, this implies that, for the revolving door combination patterns
of the form atat−1 . . . a1, if those of fixed at are considered, the maximum
and minimum ranks of such patterns are
(at+1
t
)
− 1 and
(at
t
)
respectively.
Let Rank(atat−1 . . . a1) be the rank of the pattern atat−1 . . . a1, the re-
volving door combination also has the following recursive property on its
rank,
Rank(atat−1 . . . a1) =
[(
at + 1
t
)
− 1
]
− Rank(at−1 . . . a1). (19)
As an implication of this, if all
(k
t
)
codewords need to be enumerated, for
some integers k, t > 0 and k ≥ t, we can split the enumeration into ⌈
(k
t
)
/M⌉
blocks where in each block only at most M codewords need to be enumer-
ated. In this way, the enumeration of each block can be done on a separate
computer–allowing parallelism of codeword enumeration. We know that at
the jth block, the enumeration would start from rank (j− 1)M and the cor-
responding pattern can be easily obtained by making use of (19) as well as
the maximum and minimum ranks of the patterns of fixed at.
10

Table 1: The weight distributions of [137, 69, 21] augmented and [138, 69, 22]
extended quadratic residue codes
j Q137 = [137, 69, 21] Qˆ137 = [138, 69, 22]
0 1 1
21 51238 0
22 270164 321402
23 409904 0
24 1947044 2356948
25 4057118 0
26 17476816 21533934
27 99448300 0
28 390689750 490138050
29 1445284240 0
30 5203023264 6648307504
31 18055712240 0
32 59809546795 77865259035
33 189973513945 0
34 581095454420 771068968365
35 1709208146190 0
36 4842756414205 6551964560395
37 13221982102853 0
38 34794689744350 48016671847203
39 88328700833460 0
40 216405317041977 304734017875437
41 511980845799941 0
42 1170241933257008 1682222779056949
43 2585374360137184 0
44 5523299769383984 8108674129521168
45 11414864729214318 0
46 22829729458428636 34244594187642954
47 44202380361406672 0
48 82879463177637510 127081843539044182
49 150535995889831600 0
50 264943352766103616 415479348655935216
51 451961780387038844 0
52 747475252178564242 1199437032565603086
53 1198781830242451728 0
54 1864771735932702688 3063553566175154416
55 2814110491202421488 0
56 4120661790689260036 6934772281891681524
57 5855675469990794812 0
58 8076793751711441120 13932469221702235932
59 10814690610004223000 0
60 14059097793005489900 24873788403009712900
61 17746731937729182608 0
62 21754058504313191584 39500790442042374192
63 25897686719588958304 0
64 29944200269524733039 55841886989113691343
65 33629639551783390742 0
66 36686879511036426264 70316519062819817006
67 38877142978140092004 0
68 40020588359850094710 78897731337990186714
11