1
A Construction for Binary Sequence Sets with Low Peak-to-Average Power Ratio Matthew G. Parker and Chintha Tellambura Abstract A recursive construction is provided for sequence sets which possess good Hamming Distance and low Peak-to-Average Power Ratio (PAR) under any Local Unitary Unimodular Transform (including all one and multi-dimensional Discrete Fourier Transforms). An important instance of the construction identifies an iteration and specialisation of the Maiorana-McFarland (MM) construction.
I. Introduction Bipolar Golay Complementary sequences of length 2n have a Peak-to-Average Power Ratio (PAR) ≤ 2 under the one-dimensional continuous Discrete Fourier Transform (DFT∞ 1 ) [5]. The upper PAR bound of 2 follows by forming these Complementary Sequences using Rudin-Shapiro construction [19], [20]. This set is the union of certain quadratic cosets of Reed-Muller (RM) (1, n) when expressed in Algebraic Normal Form (ANF)[4], [15]. As these sequences are a subset of RM(2, n), the Hamming Distance, D, between sequences in the set satisfies D ≥ 2n−2 . The problem of finding error-correcting codes where each codeword also has low PAR has application to Orthogonal Frequency Division Multiplexing (OFDM) communications systems [7], Multi-Code CDMA [11], [12], [17], Weight Hierarchy and Quantum Entanglement [13], [14]. However the fundamental codeset identified by Davis and Jedwab [4] (DJ sequences) suffers from vanishing rate as n increases, and much higher rates are possible and desirable, where PAR ≤ O(n) [21], [16]. A generalisation of the Rudin-Shapiro construction to other starting seeds [11], [12] allows inclusion of more low PAR quadratic cosets of RM(1, n) in the code, thereby improving code rate somewhat. Higher degree cosets can also be added, marginally increasing code rate at price of distance, D, which decreases. In this paper we present a construction for much larger codesets of sequences with tight upper bound on PAR and good distance properties. In particular, we construct bipolar sequence sets of length 2n and PAR ≤ 2t , comprising ANFs up to degree µ, where µ ≤ t for t > 1, and µ = 2 for t = 1. These codesets have PAR ≤ 2t under all Linear Unimodular Unitary Transforms (LUUTs), including one and multi-dimensional continuous DFTs [11], [12]. As LUUTs include the Walsh-Hadamard Transform (WHT) then our construction gives large codesets of (Near)-Bent functions [10], [3], [18]. These binary sequences are not just (Near)Bent but are also distant from linear sequences over all alphabets, not just over Z2 - a particularly strong cryptographic attribute. In [17], Paterson notes that the Maiorana-McFarland (MM) construction [10] provides sequences with exact PAR of 1 under the Walsh-Hadamard Transform (WHT) for even numbers of binary variables, i.e. the sequences are Bent. He then increases code rate by raising the PAR upper-bound by also including many-to-one maps in addition to the one-to-one maps usually associated with MM construction. Constructions (1) and (2) of this paper can be viewed either as a recursion or specialisation of MM so that our sequence set has low PAR under all LUUTs, not just WHT. (2) significantly improves on rate compared to the DJ codeset whilst maintaining good distance properties. We also provide an explicit construction for the complete quadratic subset of (2) using the Bruhat decomposition [2], [1]. (1) provides a way of constructing low PAR error-correcting codes of any length and alphabet. We briefly mention generalisations of (1) and (2) which exploit many-to-one maps. II. The Construction PAR is a spectral measure. We therefore define the transforms over which the spectrum is to be computed: A. Definitions Definition 1: Let l = (l0 , l1 , . . . , lrn −1 ) be a length rn complex sequence. l is defined to be unimodular if |li | = |lj |, Prn −1 ∀i, j, unitary if i=0 |li |2 = 1, and r-linear if, l= =
−n
r 2 {(a0,0 , a0,1 , . . . , a0,r−1 ) ⊗ (a1,0 , a1,1 , . . . , a1,r−1 ) ⊗ . . . ⊗ (an−1,0 , an−1,1 , . . . , an−1,r−1 )} −n Nn−1 r 2 i=0 (ai,0 , ai,1 , . . . , ai,r−1 )
where ⊗ is the ’left tensor product’, such that A ⊗ (B0 , B1 , . . .) = (B0 A, B1 A, . . .). For r prime, r-linear is called linear. M.G.Parker is with the Code Theory Group, Inst. for Informatikk, Høyteknologisenteret i Bergen, University of Bergen, Bergen 5020, Norway. E-mail:
[email protected]. Web: http://www.ii.uib.no/∼matthew/, Author funded by NFR Project Number 119390/431 C. Tellambura is with the School of Computer Science and Software Engineering, Monash University, Clayton, Victoria 3168, Australia. E-mail:
[email protected]. Phone/Fax: +61 3 9905 3196/5146
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Definition 2: Let a be the length rn vector, (a0 , a1 , . . . , arn −1 ). Then an r-symmetric permutation, πr , of a is defined Pn−1 as, πr (a) = (aπr (0) , aπr (1) , . . . , aπr (rn −1) ), where πr (i) = k=0 iπ(k) rk , where π permutes Zn , and ik ∈ Zr , ∀k. Definition 3: Lr,n is the infinite set of length rn complex r-linear, unitary, unimodular sequences. Definition 4: A rn × rn r-Linear Unimodular Unitary Transform (r-LUUT) matrix L has rows taken from Lr,n such that LL† = Irn , where † means conjugate transpose, and Irn is the rn × rn identity matrix. When r is prime, r-LUUT is called LUUT. Let si be the ith element of a length rn vector, s. r-PAR(s) is computed by measuring maximum possible correlation of s with any length rn r-linear unimodular sequence, l ∈ Lr,n : Definition 5: r-PAR(s) = rn maxl (|s · l|2 ) where l ∈ Lr,n and · means ’inner product’. When r is prime, r-PAR is termed PAR [12]. A complementary sequence set [5], [12] with respect to the unitary transform, T , is defined by the property that the sum of the spectra of the sequence set, relative to T , sum to a constant. We formalise this as follows: Definition 6: The rows of an R × R′ matrix, A, form a complementary set of R sequences under the R′ × R′ unitary transform matrix, T , if q = AτiT is unitary, where τi is the ith row of T , and the rows of A are unitary. Nn−1 Definition 7: The 2n × 2n Walsh-Hadamard (WHT) and Negahadamard (NHT) Transform matrices are i=0 H, and Nn−1 1 1 1 i 1 1 2 ∞ n n √ √ , N = 2 1 −i , and i = −1. DFT1 is the set of 2 × 2 matrices, 1 −1 i=0 N, respectively, where H = 2
the union of whose rows form a subset of L2,n such that each row satisfies ai,0 = is any complex root of unity.
√1 , 2
ai,1 =
ik ω √ 2
for some fixed k, and ω
B. Construction Let N = rn , r not necessarily prime. Let Ej and Aj , 0 ≤ j < L, be a series of R × R and R × Rj+1 complex matrices, respectively, with Ej a unitary, unimodular matrix, and Aj having unitary, unimodular rows, ai,j , where R = rt , and A0 = E0 . Let the rows of Aj−1 form a complementary set of R sequences under any Rj × Rj r-LUUT. Let γj permute ZR . Then our fundamental recursive construction is: Aj is an R × Rj+1 matrix such that, ai,j = R−j/2 ((aγ −1 (0),j−1 |aγ −1 (1),j−1 | . . . , |aγ −1 (R−1),j−1 ) ⊙ (1 ⊗ ei,j )) j
j
(1)
j
where x ⊙ y = (x0 y0 , x1 y1 , . . . , xRj −1 yRj −1 ), 1 is the length Rj all-ones vector, and ′ |′ means concatenation. Theorem 1: Let s be a length N = RL row of AL−1 . Then πr (s) satisfies PAR(πr (s)) ≤ R under all N × N r-LUUTs, where πr is any r-symmetric permutation of s. Proof: Let lj and l be unitary unimodular r-linear rows of length Rj+1 and R, respectively. Let q = Aj−1 lj−1 . Then q is unitary. By Definitions 3,4,6, the rows of Aj form a complementary R-set under any r-LUUT if q′ = Aj (lj−1 ⊗ l) is PR−1 PR−1 unitary ∀lj , l. This follows if P = i=0 | k=0 (qγ −1 (k) ei,k lk )|2 = 1, for rk , ei,k and lk the kth elements of r ,ei,j and l, j
respectively. Let z = (qγ −1 (0) l0 , qγ −1 (1) l1 , . . . , qγ −1 (R−1) lR−1 )T , and Z = Ej z. Then P = 1 if Z is unitary, which follows j j j by Parseval’s Theorem if Ej is a unitary matrix, and if z is unitary. z is unitary because q is unitary and l is unitary unimodular. Any r-symmetric permutation of s is allowed because l, lj , ei,j are all r-linear. A special case of (1) occurs when r = 2, i.e. R = 2t . Let x = {x0 , x1 , . . . , xn−1 } be a set of n binary variables. Then p(x): −n Z2n → Z2 has a bipolar representation, s = 2 2 (−1)p(x) = (s0 , s1 , . . . , s2n −1 ), where si = (−1)p(x0 =i0 ,x1 =i1 ,...,xn−1 =in−1 ) , Pn−1 and i = k=0 ik 2k is a radix-2 decomposition of i, ik ∈ {0, 1}, ∀k. We further incorporate πr and (1) becomes,
s = 2 2 (−1)p , where, PL−1 PL−2 Pt−1 p = p(x) = j=0 l=0 xπ(tj+l) fl,j (xπ(t(j+1)) , xπ(t(j+1)+1) , . . . , xπ(t(j+2)−1) ) + j=0 gj (xπ(tj) , xπ(tj+1) , . . . , xπ(tj+t−1) ) (2) where n = Lt, π permutes Zn , and where fl,j : Z2t → Z2 is such that fγj : Z2t → Z2t := (f0,j , f1,j , . . . , ft−1,j ) is a P ′ l ′ permutation governed by, i′ = γj (i), where i′ = t−1 l=0 il 2 is a radix-2 decomposition, il = fl,j (i0 , i1 , . . . , it−1 ), and each γj permutes ZR . To avoid unnecessary duplications, we exclude the fγj where one or more fl,j has a ’+1’ constant offset, as these are covered by the gj , and also the cases where all fl,j are monomials, except when fγj is the identity, as these are covered by the π permutation. Corollary 1: The length N = 2n bipolar sequence, s, generated by (2), satisfies PAR(s) ≤ 2t under all N × N LUUTs. −n
Proof: Construction (2) is a case of (1) where all Ej are 2t × 2t WHTs. γ −1 of (1) essentially determines fγ , where fγ exludes the unnecessary duplications mentioned above. The Corollary follows from Theorem 1. When L = 2, (2) reduces to MM over 2t variables. 1 For L > 2, MM links successive sets of 2t variables which overlap in t variables. (2) is therefore recursive MM. Alternatively, for L even, we can consider (2) as a subset of the possible 1 Thanks
to V.Rijmen for pointing out the MM connection.
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MM codeset over n variables, where only permutations over disjoint t-variable subspaces are allowed. Theorem 2: For fixed t, let P be the codeset of p(x) of degree µ or less, generated using (2). Then, |P| 2n+1
n
≤ ≤
n
t
Γ t −1 ( t! n!(22 −t−1 ) t ) , 2t! n n t ((2t −1)!) t −1 n!(22 −t−1 ) t 2t!
µ=2 ,
(3)
µ≥2
Qt−1 where Γ = i=0 (2t − 2i ) = |GL(t, 2)|. (GL is the General Linear Group). (Only for t = 1 or L ≤ 2 is the upper bound exact). Proof: By counting arguments, for µ = 2, ln Qt t n l=1 t ( nt )!t n |P| Γ n −1 t 2 t × (2 )t ≤ × ×( ) 2n+1 t! 2 t! t
Γ with (22t)! , which is the number of permutations excluding those with a constant offset, ’+1’. For µ ≥ 2, we replace t! The Theorem follows. In Section II-D we show how to generate all degree-one fγ , via isomorphism to the General Linear Group, where the number of degree-one permutation polynomials is Γ.
C. Examples The WHT, NHT, and DFT∞ 1 are used as ’spot-checks’ in the following examples to validate the PAR upper-bound. C.1 Example 1 Let γj be the identity ∀j. Then, fl,j (xπ(t(j+1)) , xπ(t(j+1)+1) , . . . , xπ(t(j+2)−1) ) = xπ(t(j+1)+l) , and (2) becomes, L−2 t−1
p(x) =
XX
L−1
xπ(tj+l) xπ(t(j+1)+l) +
j=0 l=0
X
gj (xπ(tj) , xπ(tj+1) , . . . , xπ(tj+t−1) )
(4)
j=0
When deg(gj ) < 2, ∀j, it is well-known that s = 2 2 (−1)p(x) is Bent (PAR = 1 under the WHT) for L even [9] and (perhaps not known) that s has PAR = 2t under the WHT for L odd. In general, for any gj , s has PAR ≤ 2t under all LUUTs. For example, if L = 4, t = 3, and p(x) = x0 x3 + x1 x4 + x2 x5 + x3 x6 + x4 x7 + x5 x8 + x6 x9 + x7 x10 + x8 x11 , then s has PAR = 1.0 under WHT and NHT, and PAR = 7.09 under DFT∞ 1 . Similarly, let g0 (x0 , x1 , x2 ) = x1 x2 , −n p(x)+g0 +g1 +g2 ′ 2 has PARs 4.0, 2.0, and 7.54 under g1 (x3 , x4 , x5 ) = x3 x4 x5 , and g2 (x6 , x7 , x8 ) = 0. Then s = 2 (−1) t WHT, NHT, and DFT∞ 1 , respectively. In all cases, PAR ≤ 2 = 8.0. −n
C.2 Example 2, PAR ≤ 2.0, (t = 1) Let t = 1. We have one permutation, fγ = x, (excluding fγ = x + 1). From (2), PL−2 p(x) = j=0 xπ(j) xπ(j+1) + cj xj + d, cj , d ∈ Z 2
(5)
This is exactly the DJ set of binary quadratic cosets of RM(1, n), where n = L [4]. This set has PAR ≤ 2.0 under DFT∞ 1 [4]. Such sequences are Bent for n even [9], [18] and, in [11], [12] it is shown that such a set has PAR = 2.0 under WHT for n odd, and also, under NHT, has PAR = 1.0 for n 6= 2 mod 3 (NegaBent), and PAR = 2.0 for n = 2 mod 3. More generally the DJ set has PAR ≤ 2.0 under any LUUT [12], and this agrees with Theorem 1. For example, let p(x) = x0 x4 + x4 x1 + x1 x2 + x2 x3 + x1 + 1 . Then s has PAR = 2.0 under the WHT, NHT, and DFT∞ 1 . The DJ set, being cosets of R(2, n), forms a codeset with Hamming Distance, D ≥ 2n−2 . The rate of the DJ codeset follows increases. This is the primary drawback of this code as its rate vanishes rapidly as n increases.
n+1 ( n! 2 )2 22n
as n
C.3 Example 3, PAR ≤ 4.0, (t = 2) [4], [16], [11], [12], [18] all propose techniques for the inclusion of further quadratic cosets, so as to improve rate at the price of increased PAR. We here propose an improved rate code (although still vanishing), where PAR ≤ 4.0. To t )! achieve this we set t = 2 in (2). There are (2 2t t! = 3 valid permutation polynomials, fγ = (f0 , f1 ). These polynomials 2 2 map from Z2 → Z2 , and are taken from the set, fγ (x0 , x1 ) ∈ {(x0 , x1 ), (x0 + x1 , x1 ), (x0 , x0 + x1 )}. Substituting for fl,j and gj in (2) gives a large set of polynomials with PAR≤ 4.0 under all LUUTs. We now list, for this construction, the p(x) arising from the 3 invertible polynomials, fγ , for one ’iteration’ of (2), i.e. for L = 2, where we fix π to the identity. p(x) = x0 x2 + x1 x3 + c0 x0 x1 + c1 x2 x3 + RM(1, 4) p(x) = x0 (x2 + x3 ) + x1 x3 + c0 x0 x1 + c1 x2 x3 + RM(1, 4) p(x) = x0 x2 + x1 (x2 + x3 ) + c0 x0 x1 + c1 x2 x3 + RM(1, 4)
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where c0 , c1 ∈ Z2 . The quadratic part of each of the above is isomorphic to a distinct invertible boolean t × t matrix, where t = 2 (Section II-D), as the permutation polynomials form a group isomorphic to the General Linear Group, Qt−1 GL(t, 2), where |GL(t, 2)| = i=0 (2t − 2i ) [8]. Any two of the above quadratics are inequivalent under permutation of the four variables, e.g., p(x) = x0 x2 + x1 x3 + c0 x0 x1 + c1 x2 x3 + RM(1, 4) and p(x) = x0 (x2 + x3 ) + x1 x3 + c0 x0 x1 + c1 x2 x3 + RM(1, 4) . An upper bound on |P| is given by Theorem 2. Substituting t = 2 into (3), n−4 n |P| < n!2 2 3 2 −1 n+1 2
(6)
Exact enumeration and construction for this set remains open, due to extra ’hidden’ symmetries, ocurring when L > 2, and induced by π. We compute the exact number of quadratic coset leaders for n = 4, 6, 8, 10, and these are compared to the upper bound of (6) in Table I. They are also compared to the n! 2 quadratic coset leaders in the binary DJ codeset (Example 2). By assigning t = 2 we have a construction for a much larger codeset than the DJ codeset and TABLE I The Number of Quadratic Coset Leaders for Construction (2) when t = 2
n Theorem 2, (6),(3), |P|/2n+1 Exact Computation(2) DJ Code /2n+1 log2 (|P|/2n+1 ) log2 (Number of quadratics)
4 6 8 10 72 12960 4354560 2351462400 36 9240 4086096 2317593600 12 360 20160 1814400 6.2 13.7 22.1 31.1 6 15 28 45
with the same Hamming Distance, D = 2n−2 , but now PAR is upper-bounded by 4.0 instead of 2.0. For example, let, p(x) = x0 x2 + x1 x2 + x1 x6 + x2 x5 + x6 x3 + x6 x5 + x5 x4 + x3 x7 + x0 x1 + x5 x3 + x7 + x1 . Then s has PARs = 1.0, 2.0, and 3.43 under WHT, NHT, and DFT∞ 1 , respectively. C.4 Example 4, PAR ≤ 8.0, (t = 3) t
)! 3 3 There are now (2 2t t! = 840 valid permutations, fγ = (f0 , f1 , f2 ). These polynomials map Z2 → Z2 . Moreover, 168 3 3 3 2 (2 − 1)(2 − 2)(2 − 2 )/t! = 6 = 28 of the polynomials are degree-one permutations leading to quadratic forms, p(x), and can be represented by the following 7 permutations.
fγ (x0 , x1 , x2 ) ∈
{(x0 , x1 , x2 ), (x0 + x2 , x1 , x2 ), (x0 + x2 , x1 + x2 , x2 ), (x0 + x1 + x2 , x1 , x2 ), (x0 + x1 , x1 + x2 , x2 ), (x0 + x1 + x2 , x1 + x2 , x2 ), (x0 + x2 , x1 + x0 , x2 + x0 + x1 )}
Substituting for fl,j and gj in (2) gives a large set of polynomials with PAR≤ 8.0 under all LUUTs. We now list, for this construction, all quadratic p(x) arising from the 7 inequivalent degree-one permutations, fγ , for one ’iteration’ of (2), i.e. for L = 2, where π is fixed as the identity: p(x) = x0 x3 + x1 x4 + x2 x5 + g(x) p(x) = x0 x3 + x0 x5 + x1 x4 + x2 x5 + g(x) p(x) = x0 x3 + x0 x5 + x1 x4 + x1 x5 + x2 x5 + g(x) p(x) = x0 x3 + x0 x4 + x0 x5 + x1 x4 + x2 x5 + g(x) p(x) = x0 x3 + x0 x4 + x1 x4 + x1 x5 + x2 x5 + g(x) p(x) = x0 x3 + x0 x4 + x0 x5 + x1 x4 + x1 x5 + x2 x5 + g(x) p(x) = x0 x3 + x0 x5 + x1 x3 + x1 x4 + x2 x3 + x2 x4 + x2 x5 + g(x)
where g(x) = c0 x0 x1 + c1 x0 x2 + c2 x1 x2 + c3 x0 x1 x2 + c4 x3 x4 + c5 x3 x5 + c6 x4 x5 + c7 x3 x4 x5 + RM(1, 6), c0 , c1 , . . . , c7 ∈ Z2 . An upper bound to |P| can be computed from Theorem 2, (3), with µ = 2, and the upper bound is compared to the total number of quadratics in n binary variables in Table II. As with t = 2, exact enumeration and construction for TABLE II The Number of Quadratic Coset Leaders for Construction (2) when t = 3
n 6 9 12 15 Theorem 2, (3), log2 (|P|/2n+1 ) 16.7 33.5 51.7 70.9 log2 (Number of quadratics) 15 36 66 105 this set remains open, due to extra ’hidden’ symmetries. By assigning t = 3 we have a construction for a codeset with Hamming Distance, D ≥ 2n−2 and PAR ≤ 8.0 under all LUUTs.
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For t = 3 we can also include cubic forms in Construction (2). There are 5040−168 = 812 degree 2 permutation 6 polynomials, fγ = (f0 , f1 , f2 ), that map from Z23 → Z23 , and lead to cubic forms, p(x). This set can be represented by 147 degree 2 permutation polynomials which are inequivalent under variable permutation. (Along with the 7 inequivalent degree 1 permutations, this makes a total of 154 inequivalent permutation polynomials for t = 3 [6], [22]). Substituting for fl,j and gj in (2) gives a large set of polynomials with PAR≤ 8.0 under all LUUTs, and Hamming Distance, D ≥ 2n−3 . An upper bound to |P| can be computed from Theorem 2, (3) with µ = 3, and the upper bound is compared to the total number of quadratics and cubics in n binary variables in Table III. Here is an example from this codeset, where TABLE III The Number of Cubic and Quadratic Coset Leaders for Construction (2) when t = 3
n Theorem 2, (3), log2 (|P|/2n+1 ) log2 (Number of quadratics and cubics)
6 9 12 15 23.6 46.3 70.4 95.5 35 120 286 560
ijk, uv is short for xi xj xk + xu xv . Let, p(x) =
034, 035, 045, 135, 145, 234, 235, 245, 367, 368, 378, 567, 568, 69A, 79A, 7AB, 89A, 345, 9AB, 03, 05, 14, 24, 25, 36, 38, 47, 58, 69, 6A, 6B, 7A, 7B, 89, 8B, 67, 78, AB
then s has PARs 4.0, 6.625, and 7.66 under the WHT, NHT, and DFT∞ 1 , respectively. In all cases, PAR ≤ 8.0. D. A Matrix Construction for all Quadratic Codes from (2) Each degree-one permutation polynomial, fγ from Z2t → Z2t can be viewed as a t × t binary adjacency matrix. Let x = {x0 , x1 , . . . , xt−1 }. We write, M ⇔ fγ (x) = (f0 (x), f1 (x), . . . , ft−1 (x)), mi,l = 1 if xi ∈ fl (x) mi,l = 0
M = {mi,l }, deg(fl (x)) = 1, and otherwise
The mapping is an isomorphism from degree-one permutation polynomials to the General Linear Group, G = GL(t, 2), of all binary t × t invertible matrices [8]. To construct all quadratics, p(x), for a given n and t we need to construct all degree one permutations, fγ . These can, in turn be constructed by generating all of G = GL(t, 2), as follows [1], [2]: Definition 8: A binary t × t ’transvection’ matrix, Xab , satisfies, Xab = {ui,j },
where ui,j = 1,
i = j, and i = a, j = b
ui,j = 0,
otherwise
Definition 9: The Borel subgroup of G over Z2 is the t × t upper-triangular binary matrices, B. Definition 10: The Weyl subgroup of G is the t × t permutation matrices, W . t
2 matrices, Xab , a < b. Let w ∈ W be a permutation of Zt and its associated t× t Assign a fixed ordering, O, to the 2 permutation matrix. For each w, form the matrix product, Xw , comprising all Xab which satisfy a < b = w(a) > w(b), where the Xab in X are ordered according to O. Theorem 3: [1], [2] (’Bruhat Decomposition’) G = Xw′ W B (7)
where Xw′ is any sub-product of Xw that maintains the ordering of the Xab matrices in Xw . Q t i All quadratic constructions using (2) can be uniquely constructed using Theorem 3, where |G| = Γ = t−1 i=0 (2 − 2 ). III. Generalisations
Lemma 20 of [17] extends MM to a large codeset with near-Bent properties, where a 1-1 map is replaced by a 2δ -1 map. We now apply a similar generalisation to (1) and (2), (proofs omitted). Pj tmax , tmax = Let N = rn , r not necessarily prime, with n = ∆L−1 , ∆j = i=0 ti , 0 ≤ j < L, where R = r ∆j tj tj complex matrices, respectively, with Ej max(t0 , t1 , . . . , tL−1 ). Let Ej and Aj be a series of r × r and R × r a unitary matrix with unimodular, r-linear rows, ei,j , Aj having unitary, unimodular rows, ai,j , and A0 such that ai,0 = ei mod 2t0 ,0 , 0 ≤ i < R. Let the rows of Aj−1 form a complementary R-set under any r∆j−1 × r∆j−1 r-LUUT. t
t
Let γj : Zrj−1 ← Zrj be a r
tj−1 tj
tj
-1 map if tj ≤ tj−1 , and a 1-r tj−1 map if tj−1 ≤ tj . Then our construction is:
Aj is an R × r∆j matrix such that, ai,j = r−∆j−1 /2 ((ahi ,j−1 |ahi+1 ,j−1 | . . . , |ah
,j−1 t i+r j −1
) ⊙ (1 ⊗ ei mod
rtj ,j ))
(8)
6 i ⌋rtj−1 , and 1 is the length r∆j−1 all-ones vector, 0 ≤ i < R. where hi = γj−1 (i mod rtj−1 ) + ⌊ rtj−1 Theorem 4: Let s be a length N row of AL−1 , as constructed by (8). Then πr (s) satisfies PAR(πr (s)) ≤ R under all N × N r-LUUTs, where πr is any r-symmetric permutation of s. The generalisation of (2) follows: −n s= 2 2 (−1)p, where , PL−2 Ptj −1 p = p(x) = j=0 l=0 xπ(∆j−1 +l) fl,j (xπ(∆j ) , xπ(∆j +1) , . . . , xπ(∆j +tj+1 −1) ) P L−1 + j=0 gj (xπ(∆j−1 ) , xπ(∆j−1 +1) , . . . , xπ(∆j−1 +tj −1) )
t
t
(9)
t
where n = ∆j , π permutes Zn , and where fl,j : Z2j+1 → Z2 is such that fγj : Z2j ← Z2j+1 := (f0,j , f1,j , . . . , ftj −1,j ) is γj in (8). Corollary 2: The length N bipolar sequence s satisfies PAR(s) ≤ 2tmax under all N × N LUUTs, where s is generated using construction (9). IV. Discussion and Open Problems We presented a construction for low PAR error-correcting codes which significantly generalises the fundamental codeset of Davis and Jedwab. An important subcase can be viewed either as recursion or specialisation of the MaioranaMcFarland construction. Construction (2) only provides a unique, implementable encoder if we can provide algorithms to generate all permutation polynomials, fγ , of degree µ − 1. µ = 1 is trivial. Section II-D provides an answer for µ = 2. But the problem is open for µ > 2. Given an algorithm to generate all permutation polynomials, then construction (2) only generates distinct p(x) for t = 1. For t > 1, π induces extra symmetries causing many p(x) to be generated more than once. This situation is reflected in (3), which is a strict upper bound for t > 1. It remains open to provide an algorithm to generate all distinct p(x). Such an algorithm would replace (3) with an exact expression. More research is then required to enumerate and provide explicit algorithms for Constructions (8) and (9), to further improve rate. It would also be interesting to choose Ej in (1) and (8) other than WHTs. One way to improve rate is to choose rectangular Ej whose rows form a set of near-orthogonal sequences. This would result in a slowly rising PAR bound as L increases, but the rate of the code would also improve. 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