Constructions of complementary sequences from 2-level - - PowerPoint PPT Presentation

Constructions of complementary sequences from 2-level autocorrelation sequences and permutation polynomials

Guang Gong

Department of Electrical and Computer Engineering University of Waterloo CANADA <https://uwaterloo.ca/scholar/ggong>

SEquences and Their Applications 2020 (SETA 2020) Virtual Conference, Sep 22 - 25, 2020, Russia, Saint-Petersburg

Joint work with Zilong Wang

• G. Gong (UW)

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Outline

Basic definitions of sequences and correlation Complementary sequences sets (CSS) and complete mutually

• rthogonal CSS (or shorted as complete complementary code (CCC))

Paraunitary (PU) matrices and CSS/CCC Explicit algebraic normal forms of CSS/CCC from DFT Hadamard matrices Sequences with idea 2-level autocorrelation and cyclic Hadamard matrices Explicit algebraic normal forms of constructed CSS/CCC from 2-level autocorrelation sequences

• G. Gong (UW)

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Historic development of Golay sequences

There are a large volume of the publications for those work, which are not shown in the above figure. Please refer to those in our earlier paper (Wang-Ma-Gong-Xue 2019).

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Sequences, periodic and aperiodic correlation

q-ary sequence of length L (or period L) is given by a = (a(0), a(1), · · · , a(L − 1)), a(t) ∈ Zq Let ω be a primitive qth root of unity, a periodic autocorrelation function at shift τ is defined as R(τ) =

L−1

• i=0

ωa(i+τ)−a(i), τ = 0, ±1, ±2, · · · where i + τ is reduced by modulo L. So, R(τ) is also periodic with period L.

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Example

An m-sequence of period 7 generated by the characteristic polynomial w(x) = x3 + x + 1 is a = 1110100 This can be generated by an LFSR with 3 stages 1 Continuous-time signal of the m-sequence:

t sc(t) +1 −1 Tc NTc

The autocorrelation has all out-of-phase autocorrelation values to equal −1.

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Remarks

Perfect sequences: all out-of-phase periodic autocorrelation is equal to zero! For binary case, only known sequence is a = 0111. For an m-sequence with period L = 2n − 1, we have the normalized autocorrelation: R(τ) L = 1 τ ≡ 0 mod L − 1

L → 0

τ ≡ 0 mod L (n → large) This property of m-sequences resembles white Gaussian noise, which makes it so popular in digital communications. After Solomon Golomb successfully used for detecting returning signal from the first satellite launched by US in 1958, it finds many applications such as detection

• f transmitted signals, GPS, radar distance range, spread spectrum, hardware

testing, etc.

• G. Gong (UW)

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Aperiodic autocorrelation

An aperiodic autocorrelation function at shift τ is defined as

L−1−τ

• i=0

ωa(i+τ)−a(i), −L < τ < L. This is the functionality of matched filters in signal process with applications in wireless communication.

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Example

We take q = 4, N = 4, a = (0, 1, 0, 3) = ⇒ ω = i, i = √ −1, {ωk| k = 0, 1, , 2, 3} = {1, i, −1, −i} Aperiodic autocorrelation computation at the fashion of matched filter detection: −1 −3 τ C(τ) 1 3 1 3 −3 −i 1 3 −2 1 3 −1 −i 1 3 4 (matched) 1 3 1 i 1 3 2 1 3 3 i

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Sequences and generating functions

The given sequence can be evaluated from a given generalized boolean function (GBF) in two variables: f(x0, x1) = 2x0x1 + x0, xi ∈ {0, 1} The truth table of f yields a: x1 x0 f(x0, x1) 1 1 1 1 1 3 So a = (f(0), f(1), f(2), f(3)) where t = x0 + x12, xi ∈ F2 is the binary representation of the integer t. Now we can associate the sequence a with its Z transform or generating function: F(Z) =

3

• t=0

if(t)Zt = 1 + iZ + Z2 − iZ3.

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Aperiodic correlation and generating function

Suppose that Z is a complex variable. The squared magnitude of F(Z) is |F(Z)|2 = F(Z)F(Z) = L +

L−1

• τ=1

Ca(τ)Zτ +

L−1

• τ=1

Ca(τ)Z−τ, where Ca(τ) is aperiodic autocorrelation of sequence a at shift τ.

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Golay Complementary Pairs

A pair of sequences (a, b) of length L is called a Golay complementary pair if Ca(τ) + Cb(τ) = 0, 0 < τ < L. Each sequence of such a pair is called a Golay sequence. If (a, b) form a Golay pair of length L, and let G(Z) be b’s generating function, then |F(Z)|2 + |G(Z)|2 = 2L In other words, if a is a Golay sequence, then |F(Z)|2 2L for all |Z| = 1, which leads to PMEPR(a) 2 . This property of Golay sequences can be used in OFDM transmission systems for reducing the peak to average power ratio.

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Sequences and arrays

For the previous example, f = (f(0), f(1).f(2), f(3)) = (0, 1, 0, 3), it is evaluated from a GBF f(y0, y1) = 2y0y1 + y0, f(t) = f(t0, t1) where t = t0 + t12, ti ∈ Z2, a binary representation of an integer t. f(y0, y1) is referred to as an array. We can associate it with a (multivariable) generating function: f(z0, z1) =

• (y0,y1)∈F2

2

if(y0,y2)zy0

0 zy1 1

= 1 + iz0 + z1 − iz0z1

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Relationship of sequences and arrays

A q-ary sequence f of length L

Corresponding function: f(t) : ZL → Zq Generating function: F(Z) =

L−1

• t=0

ωf(t)Zt

An m-dimensional q-ary array of size p × p × · · · × p

Corresponding function: f(y) = f(y0, y1, · · · , ym−1) : Zm

p → Zq

Generating function: F(z) =

p−1

• y0=0

p−1

• y1=0

· · ·

p−1

• ym−1=0

ωf(y)zy0

0 zy1 1 · · · z ym−1 m−1

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Aperiodic correlation of arrays

We can also define the aperiodic cross-correlation of two arrays f1(y) and f2(y) from Zm

p to Zq at shift τ = (τ0, τ1, · · · , τm−1) by

Cf1,f2(τ) =

• y∈Zm

p

ωf1(y+τ)−f2(y) where “y + τ”is the element-wise addition of vectors over Z, and ωf1(y+τ)−f2(y) = 0 if f1(y + τ) or f2(y) is not defined. The aperiodic autocorrelation of array f at shift τ is Cf(τ) = Cf,f(τ). Array correlation is a stronger condition than a sequence correlation.

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Example on array correlation

For previous example f(y0, y1) = 2y0y1 + y0, the array autocorrelation of f(z0, z1) Cf(τ0, τ1) =

• y∈Z2

2

if((y0,y1)+(τ0,τ1))−f(y0,y1)

y1 y0 f(y0,y1) f(y0+τ0,y1+τ1) (τ0,τ1)=(−1,−1)

Cf(−1. − 1) = if(0,0)−f(1,1) = −i

y1 y0 f(y0,y1) f(y0+τ0,y1+τ1) (τ0,τ1)=(−1,0)

Cf(−1.0) = if(0,0)−f(1,0) + if(1,0)−f(1,1) = if(0)−f(2) + if(1)−f(3) = 0

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(Cont.)

The autocorrelation of array f(y0, y1) is the sum of the overlapping points of two squares. Thus the autocorrelation of the sequence (0, 1, 0, 3) can be computed by the corresponding array correlation.                          Ca(−3) =Cf(−1, −1), Ca(−2) =Cf(0, −1), Ca(−1) =Cf(−1, 0) + Cf(1, −1), Ca(0) =Cf(0, 0), Ca(1) =Cf(1, 0) + Cf(−1, 1), Ca(2) =Cf(0, 1), Ca(3) =Cf(1, 1).

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Complementary sequence sets (CSS)

A set of sequences S = {f0, f1, · · · , fN−1} is called a complementary sequence set (CSS) of size N if the sum of their autocorrelation values at τ is equal to zero at all τ = 0, i.e.,

N−1

• j=0

Cfj(τ) = 0 for τ = 0. Equivalently using their generating functions:

N−1

• j=0

Fj(Z)F j(Z−1) = NL.

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Complete complementary codes

Mutually orthogonal CSSs: Two CSSs S1 = {f1,0, f1,1, · · · , f1,N−1} S2 = {f2,0, f2,1, · · · , f2,N−1} are said to be mutually orthogonal if

N−1

• j=0

Cf1,j,f2,j(τ) = 0, ∀τ. Equivalently,

N−1

• j=0

F1,j(Z)F 2,j(Z−1) = 0. CCC: Let Si = {fi,0, fi,1, · · · , fi,N−1} be CSSs of size N for 0 ≤ i < N. If any of two sets are mutually orthogonal, then the collection of Si is called complete mutually orthogonal complementary sets (CMOCS) or complete complementary codes (CCC).

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Matrix Representation of CCC

S is a CCC if and only if every row is a CSS of size N, and any two rows are mutually orthogonal. S =      f0,0 f0,1 · · · f0,N−1 f1,0 f1,1 · · · f1,N−1 . . . . . . ... . . . fN−1,0 fN−1,1 · · · fN−1,N−1      . Generating (polynomial) matrix: M(Z) =      F0,0(Z) F0,1(Z) . . . F0,N−1(Z) F1,0(Z) F1,1(Z) . . . F1,N−1(Z) . . . . . . ... . . . FN−1,0(Z) FN−1,1(Z) . . . FN−1,N−1(Z)      .

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CASs and CCAs

In terms of correlation of arrays, we have analogue definitions on

◮ a complementary array set (CAS) if the sum of their autocorrelation

• f N arrays is equal to zero, and

◮ a complete complementary array (CCA) if any two of N CASs are

mutually orthogonal.

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Example of CCC and CCA for q = 4, N = 2 and L = 4

CCC S = f0,0 = (0, 1, 0, 3) f0,1 = (0, 1, 2, 1) f1,0 = (0, 3, 0, 1) f1,1 = (0, 3, 2, 3)

• M(Z) =

1 + iZ + Z2 − iZ3 1 + iZ − Z2 + iZ3 1 − iZ + Z2 + iZ3 1 − iZ − Z2 − iZ3

• In other words, each row is a Golay pair and two rows are mutually orthogonal.

Corresponding CCA:

• M(x0, x1) =

2x0x1 + x0 2x0x1 + x0 + 2x1 2x0x1 + 3x0 2x0x1 + 3x0 + 2x1

• M(z0, z1) =

1 + iz0 + z1 − iz0z1 1 + iz0 − z1 + iz0z1 1 − iz0 + z1 + iz0z1 1 − iz0 − z1 − iz0z1

• In general, if we have a CCA, then its evaluation gives a CCC, but the other

direction is not TRUE.

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Standard Golay Sequences (Davis-Jedwab (1999))

Davis and Jedwab (1999) showed that q-ary Golay sequences of length 2m can be represented by an explicit algebraic normal form. a(x1, x2, · · · , xm) = q 2

m−1

• i=1

xπ(i)xπ(i+1) +

m

• i=1

cixi + c0, b(x1, x2, · · · , xm) = a(x1, x2, · · · , xm) + q 2xπ(m), where ci ∈ Zq, and π is a permutation of symbols {1, 2, · · · , m}. Issue for practical applications: The code rate is low, which stimulated tremendous amount of research to increase the code rate.

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Our approach: paraunitary (PU) matrices

A generating matrix is a paraunitary matrix (PU) if M(Z) · M†(Z−1) = c · IN. CCA ⇔ M(Z) is a PU matrix which is a generating matrix of a function matrix, referred to as a desired PU matrix. So, from this way, we can get a CCC.

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The Main challenge

Could we construct CSSs and CCCs by PU matrices? First, we need to make sure that PU matrices map to q-ary sequences. This can be done using Butson type Hadamard matrices.

Challenge

How to represent the q-ary sequences from their generating functions and extract their explicit algebraic normal forms? Studying arrays instead of sequences may reduce the difficulty of the above problem.

“Although Golay arrays have been previously studied by Luke and especially Dymond, and shown to be of use in coded imaging, it appears that for the most part they have been ignored or else regarded as merely another generalization of a familiar combinatorial object.”

• F. Fiedler, J. Jedwab and M. G. Parker, “A multi-dimensional approach to the construction

and enumeration of Golay complementary sequences,” J. Combin. Theory (Series A), vol. 115, no. 5, pp. 753–776, 2008.

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Two ingredients: Butson-type Hadamard matrices and delay matrices

A complex Hadamard matrix is a complex N × N matrix H satisfying |Hij| = 1 (i, j = 1, 2, · · · , N) and HH† = N · IN. A complex Hadamard matrix of size N is called a Butson type H(q, N) if all the entries of H are qth roots of unity. Delay matrix D(z) is a p by p diagonal matrix with the form D(z) =          z0 · · · z1 · · · z2 · · · z3 · · · . . . . . . . . . . . . ... . . . · · · zp−1          .

• G. Gong (UW)

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Seed PU Matrices

How to construct desired PU matrices?

Let H{k}s be BH matrices and M(z) = H{0} · D(z0) · H{1} · D(z1) · · · H{m−1} · D(zm−1) · H{m}. Then M(z) is a desired PU matrix, i.e., (1) M(z) is PU: M(z) · M †(z−1) = N m+1 · IN; (2) Each entry of M(z) can be expressed by the generating function of an array f(y): Zm

p → Zq.

In this way, CSSs/CCCs are obtained. But, how to extract function f(y)? Define its phase matrix H of H ∈ H(q, N) by Hi,j = s if Hi,j = ωs. The sequences are given by the phase matrix (recall the definition of correlation). Phase matrix H is the function matrix of the generating matrix H.

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Examples of representative of BH matrices

S

H(q = Even, N = 2):

H = 1 1 1 −1

• ⇒

H = q/2

• S

H(q = 3, N = 3):

H =   1 1 1 1 ω1 ω2 1 ω2 ω1   ⇒ H =   1 2 2 1   S

H(q = 2, N = 4):

H =     1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1     ⇒ H =     1 1 1 1 1 1    

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Examples of Representative of BH Matrices

S

H(q = 4, N = 4):

Walsh-Hadamard H1 =     1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1    

• H1 =

    2 2 2 2 2 2     DFT H2 =     1 1 1 1 1 √−1 −1 −√−1 1 −1 1 −1 1 −√−1 −1 √−1    

• H2 =

    1 2 3 2 2 3 2 1    

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Extracting functions from a seed PU

Given a seed PU, M(z) = H{0} · D(z0) · H{1} · D(z1) · · · H{m−1} · D(zm−1) · H{m}. in terms of the Kronecker-delta function, for each equivalent class of Hadamard matrices, we have two types of functions, referred to as δ-quadratic terms: denoted by δQ(q, N) δ-linear terms: denoted by δL(q, N)

Theorem 1 (Wang-Ma-Gong-Xue (2019))

All the functions extracted from the seed PU matrices can be represented in a general form: f(y) =

m−1

• k=1

hk(yk−1, yk) + l(y). where hk(·, ·), (1 ≤ k ≤ m − 1) are δ-quadratic terms and l(y), a δ-linear term. (resembling the standard Golay sequences!)

y0 h1(·, ·) y1 h2(·, ·) y2 h3(·, ·) ym−1 hm−1(·, ·)

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CCAs, CASs, CCCs and CSSs

Now from a seed PU matrix, we can construct all those objects. f(y) = m−1

k=1 hk(yk−1, yk)+l(y)

δ-quadratic terms: ∀h(·, ·), h′(·, ·) ∈ δQ(q, N) Permutation π: π · f = f(yπ(0), yπ(1), · · · , yπ(m−1)). CCA fu,v(y)=f(y)+h(u, y0)+h′(ym, v), u, v ∈ ZN CCC π·fu,v(y), u, v ∈ ZN CAS fu(y)=f(y)+h(u, y0), u ∈ ZN CSS π·fu(y), u ∈ ZN.

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First attempt to determine δ-linear and quadratic terms

We are able to extract δ-quadratic terms for small q and N. From those results, we obtained a large number of new CSSs/CCCs and all the known results can be explained as special cases in our constructions. Those results are presented in [1] (Wang-Ma-Gong-Xue (2019)). However, even for the case q = N = 4, the algebraic structure of the δ-quadratic terms is unclear. In the rest of the talk. we present how to determine the algebraic structure of δ-quadratic terms. [1] Z. Wang, D. Ma, G. Gong and E. Xue “New construction of complementary sequence (or array) sets and complete complementary codes, ”in the second round review of IEEE Trans. Inform. Theory. [Online]. Available: https://arxiv.org/abs/2001.04898

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Observations from the work by Wang-Ma-Gong-Xue (2019)

Any BH matrices, other than WHT, will lead to new δ-quadratic terms, which make the number of the sequences in CSSs increased exponentially. Let Nq(k) be the number of inequivalent q-ary BH matrices of order k. q-ary k Nq(k) binary 4 2 binary 16 5 binary 32 millions quaternary 8 15 quaternary 12 319 ternary 3 1 pentanary 5 1

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Deeper results for δ-linear and quadratic terms

Theorem 2

The collection of the δ-linear terms can be represented in the form: δL(q, N) = m−1

• k=0

lk(yk)

• ∀lk(yk) : ZN → Zq
• .

i.e., a δ-linear term is the sum of m functions from ZN → Zq with different variables.

Theorem 3

Let H be a BH matrix of order pn with entry Hu,v = ωh(u,v) where h(u, v) is a function from Z2

N to Zq for u, v ∈ ZN, and g(·), g′(·) be arbitrary permutation functions

• ver ZN. Then any δ-quadratic term associated with H equivalent class is given by

h(g(y0), g′(y1)) ∈ δQ(q, N), Now what the remaining task is to determine explicit algebraic normal forms of δ-quadratic terms, which are related to the algebraic structure of Hadamard matrices and permutations!

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δ-quadratic terms from DFT Hadamard matrices

DFT matrix of order N is a BH matrix with entry Hu,v = ωu·v. The entry of its phase matrix is given by

• Hu,v = u · v.

δ-quadratic terms determined by DFT matrices: g(y0)g′(y1) ∈ δQ(N, N) where g(·), g′(·) are arbitrary permutation functions over ZN. Two cases:

◮ N = p: the computation through the permutation polynomials (PP)

• ver Fp.

◮ q = N = 2n: computation through bijective GBFs from Zn

2 to Zq.

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Constructions from PPs over Fp

gk(·), g′

k(·) are arbitrary PPs over Fp

∀l(x) ∈ δL(p, p) Permutation π p-ary function f(x) =

m−1

• k=1

gk(xk−1) · g′

k(xk) + l(x).

Construction 1

(1) CSS of size p: fu(x) = f(π · x) + u · g0(xπ(0)) for u ∈ Fp. (2) CCC: fu,v(x) = f(π · x) + u · g0(xπ(0)) + v · g′

0(xπ(m−1)) for u, v ∈ Fp

For p = 2, there is only one identity PP in F2. So f(x0, · · · , xm−1) =

m−1

• k=1

xk−1xk +

• k

ckxk + c We get Golay sequences in a trivial way!

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Example for p = 5

Semi-normalized PP: g(x) is a monic PP with g(0) = 0. S(5) = {x, x3, x3 + 3x2 + 3x, x3 + x2 + 2x, x3 + 4x2 + 2x, x3 + 2x2 + 3x}. Sequence: f(x) =

m−1

• k=1

dk · gk(xπ(k−1)) · g′

k(xπ(k)) + m−1

• k=0
• ck,4x4

k + ck,3x3 k + ck,2x2 k + ck,1xk

• + c′,

CSS:                f(x), f(x) + g0(xπ(0)), f(x) + 2g0(xπ(0)), f(x) + 3g0(xπ(0)), f(x) + 4g0(xπ(0)) CCC: f(x) · J5 + g0(xπ(0)) ·       1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4       + g′

0(xπ(m−1)) ·

      1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4      

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Constructions from bijective GBFs (N = q = 2n)

Permutation function g(y) over Z2n ⇒ bijective GBF g(x) from Fn

2 to Z2n

Example There are 3 bijective GBFs from F2

2 to Z4.

     g1(x0, x1) = x0 + 2x1, g2(x0, x1) = 2x0 + x1, g3(x0, x1) = 2x0x1 + x0 + 3x1. Then there are 18 δ-quadratic terms determined by DFT matrices of order 4, which can be presented by dg(x0, x1)g′(x2, x3) for g, g′ ∈ {g1, g2, g3}, d = 1, 3. The collection of these δ-quadratic terms is denoted by δ(DF T )

Q

(4, 4).

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Construction of Hadamard matrices from 2-level autocorrelation (AC) sequences

Let α be a primitive element in Fpn. s = {s(t)}: p-ary 2-level autocorrelation sequences of length pn − 1 and s(t) = h(αt) where h(y) =

r Tr(βryr) is the trace representation of s, i.e., h(y) maps from

Fpn to Fp. From {s(t)}, a BH matrix of order N = pn, say H = (Hij), can be constructed: Hi+1,j+1 = ωs(i+j), 0 ≤ i, j < pn − 1, H0,j = Hi,0 = 1, 0 ≤ i, j < pn, where i + j is the summation over ZN−1. In the equivalence relation of Hadamard matrices, the entries of H can be represented by Hu,v = ωh(u·v) for u, v ∈ Fpn.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 38 / 57

Known constructions of 2-level AC sequences

Binary case of period 2n − 1 n Constructions Comments 2n − 1 prime quadratic residue se- quences 1932 2n − 1 = 4a2 + 27, prime Hall’s sextic residue se- quences Hall (1959) n ≥ 2 m-sequences Singer (1938), Golomb (1954) n ≥ 6, n compos- ite GMW sequences Goldon-Mills-Welch (1962), Scholtz and Welch (1984) n ≥ 7 Hyper-oval construc- tion Segre, Glynn I and II cases, Maschietti (1998) n ≥ 7 Dillon-Dobbertin’s Kasami power func- tion constructon Dillon-Dobbertin (2004), includ- ing conjectured 3-term (1997), 5-term sequences and WG se- quences (1998) as subclasses. All the known construction on binary 2-level autocorrelation sequences are collected in Golomb-Gong’s book: Signal design for good correlation, Cambridge Press, 2005.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 39 / 57

Known constructions of 2-level AC sequences (cont.)

Nonbinary case of period pn − 1 n p Constructions Comments n > 1 p > 2 m-sequences Zieler (1959) n composite GMW sequences Golden-Miller-Welch (1962) n > 1 p > 2 HG sequences Helleseth-Gong (2001) n > 2 p = 3 Two-term se- quences Helleseth-Kumar-Martin (2001), Arasu (2003) n > 1 p = 3 Lin conjectured se- quences Hu et al. (2014), Arasu et al. (2015) n > 2 p = 3 Ludkovski-Gong conjectured se- quences Ludkovski-Gong (2000), some proved by Arasu et al. (2015)

• Remark. For both binary and nonbinary cases, when n is composite, we have the

secondary construction using the subfield constructions, analogue to the GMW construction.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 40 / 57

δ-quadratic terms from 2-level AC sequences

Now we have an unexpected new construction for CSSs/CCC using δ-quadratic terms from the sequences with 2-level autocorrelation. Recall h(y) is a trace representation of a 2-level autocorrelation sequence s, over Fp of period pn − 1. h(λy) gives a shift of the sequence s. So, we can treat h(λy) as a function of two variables, λ = y0 and y = y1. δ-quadratic terms can be represented by h(g(y0) · g′(y1)) ∈ δQ(q = p, pn), where g(·), g′(·) are arbitrary PPs over Fpn.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 41 / 57

δ-quadratic terms from m-sequences

{s(t)} is an m sequence of period pn − 1. h(y) from Fpn to Fp: s(t) = h(αt) Trace representation: h(y) = Tr(y) BH matrix H: Hu,v = ωT r(u·v) δ-quadratic terms from m sequence: Tr(g(y0) · g′(y1)) ∈ δQ(q = p, pn).

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 42 / 57

Construction from 2-level AC sequences

hk(y), the trace representation of 2-level AC sequences over Fp of period pn − 1 gk(·), g′

k(·) are arbitrary PPs over Fpn

∀l(x) ∈ δL(pn, pn) Permutation π p-ary function f(y) =

m−1

• k=1

hk(gk(yk−1) · g′

k(yk) + l(y)), yk ∈ Fpn.

Construction 2

(1) p-ary CSS of size pn: fu(y) = f(y) + u · g0(y0) for u ∈ Fpn. (2) CCC: fu,v(x) = f(x) + u · g0(x0) + v · g′

0(xm−1) for u, v ∈ Fpn

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 43 / 57

Example: δ-quadratic forms from m-sequences of degree 2

For n = 2, q = 4, and F4 = {0, 1, α, α2 = α + 1}. We have two semi-normalized PPs: g1(y) = y and g2(y) = y2 in F22. Using those two PPs, we can obtain 6 δ-quadratics determined by WHT (i.e., m-sequence of degree 2). By setting y0 = x0 + x1α, y1 = x2 + x3α, we have                        Tr(y0y1) = x1x3 + x0x3 + x1x2, Tr(y0y2

1) = x0x3 + x1x2,

Tr(αy0y1) = x0x2 + x0x3 + x1x2, Tr(αy0y2

1) = x0x2 + x1x3 + x1x2,

Tr(α2y0y2

1) = x0x2 + x0x3 + x1x3,

Tr(α2y0y1) = x1x3 + x0x2. Denoted by δ(W HT )

Q

(4, 4).

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 44 / 57

δ-quadratic terms from BH matrices of order 4

There are two equivalent classes of the quaternary BH matrix of order 4. One class is equivalent to the DFT matrix of order 4, which gives 18 δ-distinct quadratic terms, as shown before. The other class is equivalent to the WHT matrix of order 4 (i.e., from m-sequences of period 3), which produces 6 δ-distinct quadratic terms. Thus, there are a total of 24 δ-distinct quadratic terms: δQ(4, 4) = δ(DF T )

Q

(4, 4) ∪ δ(W HT )

Q

(4, 4)

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 45 / 57

Example of the construction using PU matrices of order 4

We are now able to show a complete example of our constructions for a quaternary CSS

• f length 64 of size 4.

PU of order 4 H0 · D(z0) · H1 · D(z1) · H2 · D(z2) · H3 BH matrices and δ-quadratic forms

H0

• H1
• H2
• H3
• :

H0, H2 ∼ W HT

• H1, H3 ∼ DF T
• g0(u, y0)

g1(y0, y1) g2(y1, y2) g3(y2, v) : g ∈ δ(W HT )

Q

(4, 4) g ∈ δ(DF T )

Q

(4, 4)

(u, v)-th item in CCC (with general form f(x)) fu,v(x) = g0(u, y0) + g1(y0, y1) + g2(y1, y2) + l(x)

• f(x)

+g3(y2, v)

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 46 / 57

Cont.

Matrix form of CCC: f(x)·J4 + g0(0, y0)

g0(1, y0) g0(2, y0) g0(3, y0)

• ·J4 +J4 ·

g3(y2, 0)

g3(y2, 1) g3(y2, 2) g3(y2, 3)

• Selection of permutations. Suppose that we choose:

g0(x) = 2(x0x2 + x1x3) (from WHT) g1(x) = x1x3 + 3x0x3 + 2x1x2 + 2x0x2 + 2x0x1x3 (from DFT) g2(x) = 2(x0x2 + x0x3 + x1x2) (from WHT) g3(x) = x0x2 + 3x0x3 + 3x1x2 + x1x3 + 2x0x1x2+ 2x0x1x3 + 2x0x2x3 + 2x1x2x3 (from DFT) l(x) = x0x1 + 3x4x5 + x0 + 2x1 + 3x5

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 47 / 57

The sequences in CCC: each set is a CSS of size 4

             f0,0 : 0120030000320212012021222210021230131011332113232302212200320212 f0,1 : 0120030000320212123132333321132323020300221002120120030022102030 f0,2 : 0120030000320212301310111103310101202122003220300120030022102030 f0,3 : 0120030000320212230203000032203012313233110331012302212200320212              f1,0 : 0322010202300010032223202012001032111213312311212100232002300010 f1,1 : 0322010202300010103330313123112121000102201200100322010220122232 f1,2 : 0322010202300010321112131301330303222320023022320322010220122232 f1,3 : 0322010202300010210001020230223210333031130133032100232002300010              f2,0 : 0102032200100230010221002232023030311033330313012320210000100230 f2,1 : 0102032200100230121332113303130123200322223202300102032222322012 f2,2 : 0102032200100230303110331121312301022100001020120102032222322012 f2,3 : 0102032200100230232003220010201212133211112131232320210000100230              f3,0 : 0300012002120032030023022030003232331231310111032122230202120032 f3,1 : 0300012002120032101130133101110321220120203000320300012020302210 f3,2 : 0300012002120032323312311323332103002302021222100300012020302210 f3,3 : 0300012002120032212201200212221010113013132333212122230202120032

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 48 / 57

Example from 3-term sequences

{s(t)} is a binary 3-terms sequence of period 2n − 1. st = Tr(αt) + Tr(αq1t) + Tr(αq2t) where n = 2r + 1, q1 = 2r + 1 and q2 = 2r + 2r−1 + 1. Trace representation: h(y) = Tr(y + yq1 + yq2). BH matrix H: Hu,v = (−1)T r(uv+(uv)q1 +(uv)q2 ) δ-quadratic terms from the 3-term sequence: Tr(g(y0) · g′(y1) + g(y0)q1 · g′(y1)q1 + g(y0)q2 · g′(y1)q2) ∈ δQ(q, 2n), where g, g′ are permutations of F2n.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 49 / 57

• Cont. n = 5

For n = 5, we have that H with entry Hu,v = (−1)T r(uv+(uv)5+(uv)7) which is a binary BH matrix of order 32. δ-quadratic terms are: q 2Tr(g(y0) · g′(y1) + g(y0)5 · g′(y1)5 + g(y0)7 · g′(y1)7) ∈ δQ(q, 25). where g, g′ are permutations of F25.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 50 / 57

What do those functions look like?

If we set g = g′ the identity map, the boolean representation of the δ-quadratic term r(y0, y1) = q 2Tr(y0y1 + y5

0 · y5 1 + y7 0 · y7 1), y0, y1 ∈ F25

Even in this simplest case, its GBF has 10 variables, degree 6 and more than 1024 monomial terms.

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 51 / 57

New q-ary CSS of length 1024

From our construction, new q-ary CSS of length 1024 of size 32 are constructd from 3-term 2-level sequences. We write: r(x) = x + x5 + x7 for simplicity. In the following g, g′ are permutations of F25 and h, h′ are δ-quadratic terms. f(y0, y1) = Tr(r(g(y0)g′(y1)) + l(y0, y1)), yi ∈ F25 CCC fu,v(y0, y1) = f(y0, y1) + h(u, y0) + h′(y1, v), u, v ∈ F25 CSS fu(y0, y1) = f(y0, y1) + h(u, y0), u ∈ F25

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 52 / 57

The number of the sequences from our construction of 2-level AC

Estimation of the number of the binary sequences in CSSs of size 32 and length 1024. δ-linear terms: 22(25−1)+1 = 263 δ-quadratic terms from WHT (m-sequences) and 3-term sequences: each has ≈ 31! · 30!/(5!) ≈ 2214 Inequivalent BH matrices: ≈ 106 ≈ 220 Permutation π: 10! ≈ 221 Binary sequences from the primary construction: ≈ 263 · 2214 · 220 · 221 = 2318

• Remark. There are other binary sequences in CSSs of size 32 and length 1024 derived

from the secondary construction by Schmidt (2008) and Wang, et. al. (2019).

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 53 / 57

Concluding remarks and unsolved problems

Now, we can through away of the language of PU matrices by only considering the following functions: δ(y0, y1) = h(g(y0), g′(y1)), yi ∈ ZN where g, g′ are permutation functions of ZN and hij = ωh(i,j), the (i, j)th entry of a BH Hadamard matrix. Enumeration of CSSs: can we determine the exact enumeration of δ- quadratic terms? We only have the result for DFT case, but it is not solved for the case constructed 2-level autocorrelation, since permutations are inside the trace functions. δ(y0, y1) DFT 2-level AC N = p N = 2n N = pn g(y0)g′(y1) g(y0)g′(y1) h(g(y0)g′(y1)) g, g′ PPs of Fp g, g′ bijective GBFs g, g′ PPs of Fpn, h(·), the trace rep of a 2-level AC se- quence

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 54 / 57

Cont.

Since bent functions can be used to construct Hadamard matrices, what are the explicit form of quadratic terms for this class and enumeration? If it is intended to be used in OFDM for power reduction, our information rate is dramatically increased, for example, for a binary CSS with length 1024 of size 32, the PMEPR is 32, the information rate can reach 1/3. But can we have an encoding and decoding algorithms with moderate complexity which make those CSSs and CCCs practical? Can we use linear feedback shift registers to implement those CSSs and CCSs from 2-level autocorrelation sequences? The power of the PU method directly yields CCCs. Are there other new applications of CCCs, since we have so many of them (there is only one class constructed from Paterson’s work? . . .

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 55 / 57

Cont.

There are no new binary 2-level autocorrelation sequences reported since Dillon and Dobbertin’s work in 2004. Exhaustive search for those sequences have been done for n ≤ 10 before 2000. Can we settle the case n = 11 or n = 12 in this decade, since we have so many means for fast computing, even Sage is so powerful? Are all binary 2-level autocorrelation sequences (corresponding to cyclic Hadamard difference sets) known? How about nonbinary case, since we have so different structures for p = 3 and p > 3?

• G. Gong (UW)

CSS/CCC - 2-level auto - PP SETA2020 56 / 57

References

The content of the talk are presented in:

• Z. Wang and G. Gong, Z. Wang, and G. Gong, “Constructions of

complementary sequence sets and complete complementary codes by 2-level autocorrelation sequences and permutation polynomials ”[Online]. Available: CoRR abs/2005.05825, 2020. The preliminary results using the PU method is taken from: [1] Z. Wang, D. Ma, G. Gong and E. Xue “New construction of complementary sequence (or array) sets and complete complementary codes, ”in the second round review of IEEE Trans. Inform. Theory. [Online]. Available: https://arxiv.org/abs/2001.04898

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