NEW COMPLEMENTARY SETS OF LENGTH 2 m AND SIZE 4

. We construct new complementary sequence sets of size 4, using a graphical description. We explain how the construction can be seen as a special case of a less explicit array construction by Parker and Riera and, at the same time, a generalization of another construction by the same authors. Some generalizations of the construction are also given, which are not in the construction of Parker and Riera. Lower bounds and upper bounds of the number of sequences in the constructions are analyzed. 68P30; Secondary: 05A15.


Introduction
Orthogonal frequency-division multiplexing (OFDM) is a communication technique used in several wireless communication standards such as IEEE 802. 16 Mobile WiMAX. A major problem with OFDM is the large peak-to-mean envelope power ratio (PMEPR) of uncoded OFDM signals. Please refer to Litsyn's book [7] for a general source on PMEPR control.
Golay complementary sequences [6] have PMEPR less than or equal to 2.0, which is very low, and Davis and Jedwab [3] showed how to construct 'standard' 2 h -ary Golay complementary sequences of length 2 m , comprising some second-order cosets of the generalized first-order Reed-Muller codes RM 2 h (1, m). Paterson [11] extended the theory, and Chen et al [2] gave a tighter upper bound on the PMEPR of all the cosets of the generalized first Reed-Muller codes RM q (1, m). In [19], Yu and Gong proposed a method to construct near-complementary sequences, and gave several classes of near-complementary sequences with PMEPR≤ 4 by using shortened and extended Golay pairs as the seed pairs. In [16,18], the PMEPR bound of these near-complementary sequences were improved to asymptotically equivalent to 2, and new near-complementary sequences constructed by other seed pairs were also proposed.
Fiedler, Jedwab, and Parker [4] proposed a matrix framework for constructions of Golay sequences. [5,[8][9][10] show that the complementary set construction is primarily an array construction, where sequence sets are obtained by considering suitable projections of the arrays. It is desirable to propose constructions that significantly improve code rate without greatly compromising the upper bound on PMEPR or pairwise distance. See [1,17] for some recent results on such constructions.
In this paper, we propose constructions for complementary 4-sets instead of complementary pairs, having a very simple graphical description and with pairwise Hamming distance ≥ 2 n−2 . We show how this is a special case of the general array construction given in [8], and clarify aspects of section 5 of [10]. Our construction also generalizes a more explicit construction in [8]. Lower and upper bounds on the number of sequences generated are analyzed. Some generalizations of the construction are also given, which are not in the construction given in [8].

Preliminaries
Let ξ = e 2π √ −1/H , where H is an even positive integer. In an OFDM system with n subcarriers and H-PSK modulation, the transmitted signal for an H-ary sequence a = (a 0 , a 1 , · · · , a n−1 ) can be modeled as the real part of the complex envelope, and can be written as where f is the frequency separation between adjacent subcarriers, f 0 is the carrier frequency, and j = √ −1. Let A = (A 0 , A 1 , · · · , A n−1 ), where A i = ξ ai , and let A(z) = n−1 i=0 ξ ai z i , where z ∈ {e j2πt |0 ≤ t < 1}. Then is the aperiodic autocorrelation of a. (a,b) is a Golay complementary pair of length n if C a (τ ) + C b (τ ) = 0 for 0 < τ < n. Each sequence of a pair is called a complementary sequence. The associated polynomials, A(z), B(z), satisfy |A(z)| 2 + |B(z)| 2 = 2n, and PMEPR(a) ≤ 2 since |A(z)| 2 ≤ 2n for any z with |z| = 1.
Definition 2.1. A set of N length n sequences a 0 , a 1 , · · · , a N −1 is a complementary set if C a 0 (τ ) + C a 1 (τ ) + · · · + C a N −1 (τ ) = 0, for all τ = 0 and the associated polynomials, A 0 (z), A 1 (z), · · · , A N −1 (z), satisfy Let f : Z m 2 → Z H be a generalized Boolean function, and u = (u 0 , u 1 , · · · , u m−1 ) be the binary expansion of u, where u m−1 is the most significant bit. f (x) can be represented by its algebraic normal form (ANF): where λ u ∈ Z H . The sequence of length 2 m over Z H associated with f (x) is given by The generalized rth order Reed-Muller code RM H (r, m) of length 2 m is generated by the monomials of degree at most r. Davis and Jedwab [3] showed that 2 h -ary standard complementary sequences of length 2 m can be obtained from an explicit ANF, and Paterson [11] generalized the alphabet to an even positive integer H.
where u i , e ∈ Z H , and π is a permutation of {0, · · · , m − 1}. Then (a, b) is a Golay complementary pair, i.e. a standard pair.
where q ij ∈ Z H . Associate m-vertex labeled graph G(Q) with Q, where the edge between vertex i and j is labeled q ij .
and their compressed sequences are Golay pairs.   Proof. Let A(z), B(z), C(z), D(z) be the associated polynomials of a, b, c, d, respectively. The associated polynomial of (a + c 0 , a + c 1 , · · · , a + c m−1 ) is This completes the proof.
be a standard Golay pair of length m = 2 t . Let C and D be their complex-valued sequences, respectively. Then for any 0 ≤ i < t and r i ∈ Z 2 , is also a Golay pair, where each sequence has length 2 t+1 .
) are both Golay pairs. Thus the desired conclusion follows from Using Lemma 3.4 several times, we have the following corollary.
is also a Golay pair, each sequence has length 2 t+l .
From Lemma 3.2, we know that C | (x4x2=00) and C | (x4x2=01) is a golay pair, so is Using Lemma 3.5 several times, we have the following lemma. Lemma 3.6. Let (A, B, C, D) be a complementary set of size 4, each complexvalued sequence has length n = 2 k . Then for any r > 0 and any d ∈ Z r 2 , the complex-valued sequences sets and of length 2 r+k are both complementary sets of size 4.

Proof. (of Theorem 3.1 by induction.)
There are two initial cases: is also a complementary set of size 4. We only give the proof of the theorem for the first initial case , one can prove the theorem for the second initial case in a similar way. From Note that the end vertexes of the two paths (in this case, each path have only one vertex) are x π(0) and x π(k) , respectively. The vertexes next to x π(0) and x π(k) are x π(1) and x π(k+1) , respectively.
• Case 4: Suppose that x π(1) is connected with x π(0) , but not connected with x π(k) . This case can be proved in a similar way as in Case 3.

Suppose now that (
is a complementary set of size 4, each sequence has 2 i+2 nonzero elements, where e ∈ Z H . Note that the two end vertexes are x π(k1) and x π(k2) , where x is a list of m − (i + 2) variables. The vertexes next to x π(k1) and x π(k2) are x π(k1+1) and x π(k2+1) , respectively. Similarly, there are four cases. We only show the proof for the first case, one can prove the other three cases similarly.
Using the iteration m − 2 times, we get (A m−2 , B m−2 , C m−2 , D m−2 ) = ( H 2 Q + L, H 2 (Q + x π(k−1) ) + L, H 2 (Q + x π(m−1) ) + L, H 2 (Q + x π(k−1) + x π(m−1) ) + L) is a complementary set of size 4, where L is a linear function of m variables. Each sequence has 2 m nonzero elements, which means that they are sequences over Z H , where Q is defined in Theorem 3.1. This completes the proof. Proof. From Theorem 3.1, we know that
Since deleting the vertex 10 will obtain the quadratic form Q, as shown in Fig. 2. According to Theorem 4.1, is a Golay complementary set of size 8.
When m = 3, the number of cosets with PMEPR≤ 4 given in Corollary 2 is An upper bound on the number of cosets with PMEPR≤ 4 given in Corollary 2 is The proof of Theorem 5.1 is given in the Appendix. Table 1 shows lower and upper bounds of cosets given in Theorem 5.1 for H = 2, 4, and 3 ≤ m ≤ 8. In Table 1, if H = 2, and if H > 2, is the low bound of cosets given in [11] and is the number of cosets with PMEPR ≤ 4 obtained by two standard Golay pairs. H m−1 if H = 2 h , h ≥ 2) of cosets with PMEPR≤ 4 in [13,Construction 14] is larger than the number of cosets in Corollary 2. This is because that the sequences in [13,Construction 14] have algebraic degree 3 and effective degree 2 (see [13]), but our sequences have both algebraic degree and effective degree 2. It should be noted that lots of sequences in Corollary 2 are not in the sequences family given in [13], moreover, Corollary 2 can be applied to any even H, while in [13], H should be a power of 2, i.e., H = 2 h , where h ≥ 1.
If we set H = 2 h , then by using Theorem 4.1 and the method given in [13], we can obtain a large family of sequences with PMEPR≤ 8 and Lee weight 2 m−2 . First we introduce some definitions in [13]. Let f : {0, 1} m → Z 2 h be a generalized Boolean function. Define the effective degree of f to be Define F(r, m, h) to be the set of all generalized Boolean functions of m variables and effective degree at most r. It is shown in [13] that Thus, we have log 2 |F(0, 1, h)| = 2h − 1 and log 2 |F(1, 1, h)| = 2h. For 0 ≤ k < m, 0 ≤ r ≤ k + 1, and h ≥ 1, define the code A(k, r, m, h) to be the set of words corresponding to the set of polynomials

The number of codewords in
Generally, the number of effective degree at most r (note that where N is defined in Theorem 5.1 with m − 1 variables. For r = 1, h = 2, the efficient degree is at most 1, thus Q 0 must be equal to Q 1 , and the intermediate terms of Q 0 have coefficient H/2 = 2. For r = 2, h = 1, the efficient degree is at most 2, thus Q 0 must be equal to Q 1 . Let D be a coset of A(1, r, m, h) that contains a word in R(m, h) as the coset leader. By restricting any function corresponding to a word in A(1, r, m, h) in the variable x m−1 , we obtain an affine function, and by restricting any function corresponding to a word in R(m, h) in the same variable x m−1 , we obtain a quadratic function, whose graph is Q as defined in Corollary 2. Thus according to Theorem 4.1, the PMEPR of the coset D is at most 2 1+2 = 8. Now we are ready to give the following construction. Construction 1. Let k = 1. Let 2 ≤ r ≤ k +2 = 3 for h = 1, and 1 ≤ r ≤ k +1 = 2 for h > 1. Let r = min{r, 2}. Now take the union of N 2 min{1,r+h−3} distinct cosets of A(1, r , m, h), each containing a word in R(m, h) with effective degree at most r. The PMEPR in this code is at most 8, and from [13,Th. 9], one can show that its minimum Lee weight is at least 2 m−r . The number of words in this code is N 2 min{1,r+h−3} 2 s . Remark 2. Note that we restrict r = min{r, 2} in Construction 1. This is because that r − 1 ≤ k = 1, and to make sure that any word in A(1, r , m, h) has effective algebraic degree at most r (as a consequence, it has minimum Lee weight at least 2 m−r ), thus we have r ≤ r.
In (7), let t = 2, F −1,i (z −1 ) = 1, where 0 ≤ i ≤ 3. Let a 0 = (a 0,0 , a 1,0 ) = (1, 1). Then Let f j,r be the associated j v=0 wt(a v )-variable Boolean function of the sequence F j,r , for any j and r ≤ 2 t , where F j,r is the coefficient sequence obtained, by projection, from the polynomial F j,r (z j ), by equating z i+1 = z 2 i , ∀i. Let If O 0 = I, where I is the identity matrix then, by (7), f 0,π(0) = 0 + c 0 , f 0,π(1) = x 0 + c 1 , f 0,π(2) = x 1 + c 2 , f 0,π(3) = x 0 + x 1 + c 3 , where π is a permutation of {0, 1, 2, 3}, c i ∈ Z 2 , and π, c i are determined by P γ,0 . Similarly, for any other O 0 which have two or four −1 elements, we have, where Q is the quadratic function of j + 3-variables defined in Theorem 3.1, and x a and x b are the two end vertices of the two paths, respectively. At the end of this section, we point out that some sequences constructed in Theorem 3.1 and Corollary 2 cannot be obtained by Theorem 1 in [2]. For example, the sequence represented by Fig. 1 in Example 3 cannot be obtained by [2, Theorem 1].

Conclusion
We have given an explicit construction for complementary 4-sets, being a special case of a more general, but less explicit, construction by Parker and Riera [8]. Our construction 1 generalizes, for H = 2, construction (18) in [8]. Weak lower and upper bounds on the number of sequences generated by our construction are given, and we hope to tighten these bounds in the future work. Some generalizations of the construction are also given, which are not in the construction given in [2,8].
Let N 1 = N 1a + N 1b . Note that all the cases do not overlap with each other, so a lower bound of the number of sequences with PMEPR≤ 4 in corollary 2 is where m! 2 is the number of cosets of standard Golay sequences. Proof of the case m = 3: For m = 3, we know that there are 3 quadratic terms, and at least one of them has coefficient H 2 . Thus the number of cosets is Proof of the upper bound: For each 1 ≤ k ≤ m/2 , the number of cosets in Corollary 2 is less than Summing up all the k's satisfy 1 ≤ k ≤ m/2 , one can get the upper bound This completes the proof.