IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 11, NOVEMBER 2008 849
Complete Complementary Codes and Generalized Reed-Muller Codes
Chao-Yu Chen, Chung-Hsuan Wang, and Chi-chao Chao
Abstract—Due to ideal autocorrelation and cross-correlation properties, complete complementary codes (CCCs) can be em-ployed in CDMA systems to eliminate the multiple-access inter-ference. In this letter, we propose a direct general construction of CCCs from cosets of the first-order Reed-Muller codes, which includes previous results as a special case. The larger number of CCCs constructed by our method can provide advantages in applications to cellular CDMA systems.
Index Terms—Complete complementary codes, Golay comple-mentary sets, Reed-Muller codes, CDMA.
I. INTRODUCTION
T
HE concept of complementary pairs was first proposed by Golay [1], and then it was extended to Golay comple-mentary sets (GCSs) in [2], where the aperiodic autocorrela-tions of all the sequences in a GCS are summed to zero except at zero shift. Later in [3] the complete complementary codes (CCCs) were proposed, which can be regarded as a collec-tion of GCSs with the addicollec-tional aperiodic cross-correlacollec-tion property. Since CCCs have these autocorrelation and cross-correlation properties, they can be applied to multicarrier CDMA (MC-CDMA) systems to eliminate the multiple-access interference (MAI) [4], [5]. Other possibilities include appli-cations to radar systems and cellular OFDM systems.A connection between complementary sequences and Reed-Muller (RM) codes was first proposed in [6]. Then the rela-tionship was extended to GCSs in [7]–[9]. While a recursive construction of CCCs was provided in [10], both in [11] and this letter connections of CCCs and cosets of the first-order RM codes are proposed. However, only second-order cosets are considered in [11]; in addition, every CCC constructed in [11] contains only two distinct GCSs regardless of the order of the CCC and the other GCSs are obtained by reordering the sequences in the two distinct sets. In this letter, we provide a general relationship between CCCs and all the cosets of the first-order RM codes and can also construct CCCs consisting of all distinct GCSs. Furthermore, the construction of CCCs from second-order cosets of the first-order RM codes proposed in [11] can be regarded as a special case of our general construction. Therefore, our result can construct much more CCCs than those in [11], which can provide advantages in applications to cellular CDMA systems.
Manuscript received July 24, 2008. The associate editor coordinating the review of this letter and approving it for publication was V. Stankovic. This work was supported by the National Science Council, Taiwan, R.O.C., under Grant No. NSC 93–2213–E–007–021.
C.-Y. Chen and C.-C. Chao are with the Institute of Communications Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]).
C.-H. Wang is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/LCOMM.2008.081189
II. COMPLETECOMPLEMENTARYCODES
Let c = (c0, c1, . . . , cn−1) and d = (d0, d1, . . . , dn−1) be
Zq-valued sequences of length n, where ci and di are in the
ring Zq = {0, 1, . . . , q − 1}. In this letter, we consider the
ring Zq for even q. The aperiodic cross-correlation function
ρ(c, d; u) of c and d at displacement u is defined as
ρ(c, d; u) =
n−1−u
k=0 ξck+u−dk, 0 ≤ u ≤ n − 1
n−1+u
k=0 ξck−dk−u, −n + 1 ≤ u < 0
where ξ = e2πj/q is a primitive complex qth root of unity. We
also define the aperiodic autocorrelation function ρ(c; u) of a sequence c at displacement u to be ρ(c; u) = ρ(c, c; u).
Definition 1: [2] A set of N sequences c0, c1, . . . , cN −1
of length n is called a GCS of order N if the autocorrelation functions satisfy
ρ(c0; u) + ρ(c1; u) + · · · + ρ(cN −1; u) =
0, u= 0
N n, u = 0. Definition 2: [3] The N sets of N length-n sequences {c0
0, c01, . . . , c0N −1}, . . . , {cN −10 , cN −11 , . . . , cN −1N −1} are called
a CCC of order N if every set is a GCS and every two distinct GCSs satisfy the additional ideal cross-correlation property
N −1
k=0
ρ(cik, cjk; u) = 0, for any u; i, j = 0, . . . , N − 1; i = j.
One application of CCCs to MC-CDMA systems is to assign different GCSs in the same CCC to different users and send the composing sequences in a GCS on different carriers [5]. The receiver then correlates the signals on different carriers by respective sequences in the GCS, and hence ideal cross-correlations between different GCSs can eliminate the MAI.
III. CCCS FROMRM CODES
Let the rth-order RM code of length 2m over Z 2 be
represented by RM(r, m) and denote the 2m-tuple vectors by
vi= (00 · · · 0 2i−1 11 · · · 1 2i−1 00 · · · 0 2i−1 · · · 11 · · · 1 2i−1 ), i = 1, 2, . . . , m
and v0 = (11 · · · 1) which is the all-one vector. Note that vi
defined above has the property that the jth bit of vi is equal
to ji, where (j1, j2, . . . , jm) is the binary representation of j
with j1 the least significant bit. RM(r, m) is a binary linear code generated by the generator matrix [12] GRM(r, m) =
[vT
0, vT1, vT2, . . . , vTm, (v1v2)T, . . . , (vm−1vm)T, . . . , up to
products of degree r]T, where the product of vectors
corre-sponds to the component-wise product. The generalized rth-order RM code [7] of length 2m, denoted by RM
q(r, m),
is defined as the linear code over Zq generated by the
850 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 11, NOVEMBER 2008
same generator matrix as that of the binary RM code. It is straightforward to obtain that any codeword in RMq(r, m) can be uniquely expressed as a linear combination of rows of
GRM(r, m) over Zq with operations modulo q. The following
theorem provides a direct general construction of CCCs from cosets of the first-order RM codes, whereNm= {1, 2, . . . , m}
and v0
i = v0, v1i = vi for vi defined above.
Theorem 1: For any even integer q, any positive integer m, and any positive integer k ≤ m, let nonempty sets I1, I2, . . . , Ik be a partition of Nm, mα = |Iα|, and πα be
a bijection fromNmα to Iαfor α = 1, 2, . . . , k. Also let
Q = q 2 k α=1 mα−1 β=1 vπα(β)vπα(β+1) + k α=2 mα β=1 2α−1−1 τ =0 λα,β,τvπα(β) α−1 γ=1 vτπγγ(mγ) (1) where λα,β,τ ∈ Zq and (τ1, τ2, . . . , τα−1) is the binary
representation of τ. For any codeword c ∈ Q + RMq(1, m)
and for p, n = 0, 1, . . . , 2k− 1, if we let
cpn= c + q 2 k α=1 nαvπα(1)+ q 2 k α=1 pαvπα(mα)
where (n1, n2,· · · , nk) and (p1, p2,· · · , pk) are the binary
representations of n and p, respectively, and let Gp =
{cp
0, cp1, . . . , cp2k−1}, then G0, G1, . . . , G2 k−1
form a CCC of order 2k and length 2m.
Proof: In the first part, since ρ(d; u) = ρ∗(d; −u) for
any sequence d, to demonstrate that every Gp is a GCS of
order 2k for p = 0, 1, . . . , 2k− 1, we have to show that for
u > 0, d∈Gp 2m−1−u i=0 ξdi+u−di = 2m−1−u i=0 d∈Gp ξdi+u−di= 0.
For any integer i, let j = i + u; also let (i1, i2, . . . , im)
and (j1, j2, . . . , jm) be the binary representations of i and j,
respectively.
Case 1: If iπα(1) = jπα(1) for some α ∈ {1, 2, . . . , k},
then for any sequence d ∈ Gp, there exists d =
(d0, d1, . . . , d2m−1) = d + (q/2)vπα(1) ∈ Gp such that dj− di− dj+ di = q 2 iπα(1)− jπα(1) ≡ q 2 (mod q). So we have ξdj−di/ξdj−di = ξq/2 = −1 which implies
ξdj−di+ ξdj−di= 0. Hence, we have
d∈Gp
ξdi+u−di= 0.
Case 2: In this case, we have iπα(1) = jπα(1) for all α =
1, 2, . . . , k. Suppose that iπα(β)= jπα(β)for α = 1, 2, . . . , ˆα−
1, β = 1, 2, . . . , mα and ˆβ is the smallest integer such that
iπ
ˆ
α( ˆβ) = jπα( ˆˆ β). Let i and j be integers that are different
from i and j in only one position παˆ( ˆβ− 1), i.e., iπ
ˆ
α( ˆβ−1) =
1 − iπα( ˆˆ β−1) and jπ
ˆ
α( ˆβ−1) = 1 − jπα( ˆˆ β−1), respectively, and
so j = i+ u. For any sequence d ∈ Gp, d can be expressed
as d = Q+m
l=0glvlwhere gl∈ Zq, and then we can obtain
di− di= q 2 iπα( ˆˆ β−2)iπα( ˆˆ β−1)− iπα( ˆˆ β−2)iπα( ˆˆ β−1) +iπα( ˆˆ β−1)iπ ˆ α( ˆβ)− iπα( ˆˆ β−1)iπα( ˆˆ β) + 2α−1ˆ −1 τ =0 λα, ˆˆβ−1,τiπα( ˆˆ β−1) ˆ α−1 γ=1 iτπγγ(mγ) −2 ˆ α−1−1 τ =0 λα, ˆˆβ−1,τiπ ˆ α( ˆβ−1) ˆ α−1 γ=1 iτπγγ(mγ) + gπα( ˆˆ β−1)iπ ˆ α( ˆβ−1)− gπα( ˆˆ β−1)iπα( ˆˆ β−1) ≡q 2 iπ ˆ α( ˆβ−2)+ iπα( ˆˆ β) + gπα( ˆˆ β−1)(1 − 2iπα( ˆˆ β−1)) + 2α−1ˆ −1 τ =0 λα, ˆˆβ−1,τ 1 − 2iπα( ˆˆ β−1) α−1ˆ γ=1 iτπγγ(mγ) (mod q)
where the first equality follows from (1) and the condition that
i and i differ in only one position παˆ( ˆβ− 1). Note that we
assume ˆβ ≥ 3 here. For the case ˆβ = 2, we can just remove
those terms involving ˆβ− 2 in the preceding equation. Since iπ ˆ α( ˆβ−2) = jπα( ˆˆ β−2), iπα( ˆˆ β−1) = jπα( ˆˆ β−1), and iπγ(mγ) = jπγ(mγ) for γ = 1, 2, . . . , ˆα − 1, we have dj− di− dj+ di ≡ q 2 iπ ˆ α( ˆβ)− jπα( ˆˆ β) ≡q 2 (mod q)
which implies ξdj−di+ ξdj−di = 0. Hence, we have
d∈Gp
ξdi+u−di+ ξdi+u−di = 0.
Combining these two cases, we can obtain that Gp is a GCS
for all p = 0, 1, . . . , 2k− 1.
Then, in the second part, we will show that any two distinct sets Gs and Gt where 0 ≤ s = t ≤ 2k− 1 satisfy the ideal
cross-correlation property. For u > 0, we have to show
2k−1 n=0 2m−1−u i=0 ξcsn,i+u−ctn,i = 2m−1−u i=0 2k−1 n=0 ξcsn,i+u−ctn,i= 0 (2) where we denote cp n = (cpn,0, cpn,1, . . . , cpn,2m−1) for p = s, t
and n = 0, 1, . . . , 2k − 1. Similarly, for any integer i, let
j = i + u.
Case 1: If iπα(1) = jπα(1) for some α ∈ {1, 2, . . . , k},
then for sequences cp
n = (cpn,0, c p
n,1, . . . , c p
n,2m−1) ∈ Gp, p =
s, t, there exist sequences cpn = (cpn,0, cpn,1, . . . , cpn,2m−1) = cpn+ (q/2)vπα(1)∈ Gp such that csn,j− ctn,i− csn,j+ ctn,i=
(q/2)iπα(1)− jπα(1) ≡ q/2 (mod q). So, similar to Case
1 of the first part, we have2n=0k−1ξcsn,i+u−ctn,i = 0.
Case 2: In this case, we have iπα(1) = jπα(1) for all α =
1, 2, . . . , k. Let ˆα, ˆβ, i, and jbe given as in Case 2 of the first
part. We can also obtain that cs
n,j− ctn,i− csn,j+ ctn,i ≡ q/2
(mod q), and hence 2n=0k−1ξcsn,i+u−ctn,i+ ξcsn,i+u−ctn,i = 0.
From Cases 1 and 2, we can obtain that (2) holds for u > 0. Similarly, it can also be obtained that 2k−1
n=0
2m−1+u i=0 ξc
s
CHEN et al.: COMPLETE COMPLEMENTARY CODES AND GENERALIZED REED-MULLER CODES 851 TABLE I
COMPARISON OFΔ(k, m)ANDΔNew(k, m)FORm = 4
k 1 2 3 4 Orders of CCCs(2k) 2 4 8 16 Δ(k, 4) 12 49 63 64 Δnew(k, 4) 12 52 479 2048 to show that 2k−1 n=0 ρcsn, ctn; 0 = 2k−1 n=0 2m−1 i=0 ξcsn,i−ctn,i = 0.
For any nonnegative integer n < 2k, we have cs
n − ctn ≡
(q/2)d (mod q) where d = (s1 ⊕ t1)vπ1(m1) ⊕ (s2 ⊕
t2)vπ2(m2)⊕ · · · ⊕ (sk ⊕ tk)vπk(mk) and ⊕ denotes mod-2
addition; (s1, s2, . . . , sk) and (t1, t2, . . . , tk) are the binary
representations of s and t, respectively. It can be easily obtained that the Hamming weight of d is 2m−1. Hence,
for i = 0, 1, . . . , 2m− 1, there are 2m−1 pairs (cs n,i, ctn,i)
such that ξcsn,i−ctn,i = ξq/2 = −1 and 2m−1 pairs (cs n,i, ctn,i)
such that ξcsn,i−ct
n,i = ξ0 = 1. So we have ρ (cs
n, ctn; 0) =
2m−1 i=0 ξc
s
n,i−ctn,i= 0, which completes the proof.
Example 1: For q = 2, m = 6, and k = 3, we let I1 =
{1, 2}, I2 = {3, 4}, I3 = {5, 6}, π1(1) = 1, π1(2) = 2,
π2(1) = 3, π2(2) = 4, π3(1) = 5, and π3(2) = 6. If we denote Q = 3α=11β=1vπα(β)vπα(β+1)+ vπ3(2)
2
γ=1vπγ(2) =
v1v2+ v3v4+ v5v6+ v6v2v4, then for any codeword c in
this third-order coset Q + RM(1, 6), we have that the sets
Cp = {c + p
1v2+ p2v4+ p3v6+ n1v1+ n2v3+ n3v5 :
ni∈ Z2} for p = 0, 1, . . . , 7, where (p1, p2, p3) is the binary
representation of p, form a CCC of order 8 and length 64. Note that these 8 GCSs are all distinct.
From Theorem 1, we know that every sequence in the coset Q + RMq(1, m) lies in a CCC and hence the coset consists of
several CCCs. If mα ≥ 2 for all α = 1, 2, . . . , k, then it can
be found that our constructed CCCs of order 2kcomprises 2k
all distinct GCSs while the CCCs constructed in [11] contains only two distinct GCSs, regardless of the order 2k, and the
other GCSs are obtained by reordering the sequences in the two distinct GCSs. The coset representatives Q given in (1) can cover all the cosets of RMq(1, m) while only the second-order cosets are considered in [11]. Furthermore, if we set
mα = 1 for α = 1, 2, . . . , k − 1 in Theorem 1 and the
coset representatives Q given in (1) are restricted to second-orders, then the coset representatives Q can be reduced to those proposed in [11].
We denote the numbers of cosets of RM(1, m) which consists of CCCs of order 2k and can be derived by the
construction in [11] and our construction by Δ(k, m) and Δnew(k, m), respectively. As an example shown in Table I,
it can be found that Δnew(k, m) is much larger than Δ(k, m)
when k increases since our construction can contain high-order cosets of RM(1, m) while only second-high-order cosets are considered in [11].
Note that when CCCs are applied to cellular CDMA sys-tems, MAI can be eliminated within a cell since each user
within a cell is assigned a different GCS in the same CCC [5]. Hence in order to serve a large number of users within a cell, a CCC of a large order is desired. Furthermore, different CCCs should be employed for adjacent cells, and hence it is required that GCSs assigned in adjacent cells should have low cross-correlations to achieve low adjacent-cell interference. Since we can construct much more CCCs than those in [11], we can have more candidates of CCCs for use in adjacent cells to obtain low cross-correlations. For example, for order 8 and length 16, from our construction we can find two binary CCCs of which the largest sum of mutual cross-correlations of two different GCSs in the two CCCs over all shifts is 32, of which the largest possible value could be 8 · 162= 2048 for any two
sets of order 8 and length 16, while that of CCCs constructed in [11] is 64.
IV. CONCLUDINGREMARK
In this letter, we provide a general connection between CCCs and cosets of first-order RM codes, which includes the results in [11] as a special case. Besides applications to cellular CDMA systems, due to ideal autocorrelation/cross-correlation properties and low peak-to-average power ratios [6]–[9], one possible further application of CCCs could be to replace pseudo-noise sequences as preamble sequences in cellular OFDM systems [13].
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