Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Cheng-Yu Pai 1 and Chao-Yu Chen 2 Department of Engineering Science National Cheng Kung University Tainan 701, Taiwan, R.O.C. Email: { n98081505 1 , super 2 } @mail.ncku.edu.tw June 21-26, 2020 Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 1 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Outline Introduction Definitions Two-dimensional (2-D) Aperiodic Correlation Functions Golay Complementary Array Pair (GCAP) Golay Complementary Array Mate Generalized Boolean Functions Proposed Constructions A Construction of 2-D GCAPs from Boolean Functions A Construction of Golay Complementary Array Mate Conclusions Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 2 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Introduction The one-dimensional (1-D) Golay complementary pair (GCP) has the well-known autocorrelation property. The sum of autocorrelations of constituent sequences is zero except for the zero shift. In [1] ∗ , a connection between 1-D GCPs and generalized Boolean functions was proposed. Such GCPs are called the Golay-Davis-Jedweb (GDJ) pairs. The 2-D GCAP is an extension of 1-D GCP . The sum of 2-D autocorrelations is zero except for the 2-D zero shift. There are no explicit constructions from Boolean functions. ∗ [1] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory , vol. 45, no. 7, pp. 2397–2417, Nov. 1999 Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 3 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Introduction Most constructions of 2-D GCAPs need the aid of 1-D sequences or existing arrays. Kronecker product of 1-D GCPs and/or 2-D GCAPs [2] † . Concatenating 1-D GCPs or interleaving 2-D GCAPs [3] ‡ . Projection of three-dimensional (3-D) GCAPs [4,5] § ¶ . Based on 2-D perfect arrays [6] � . † [2] M. Dymond, “Barker arrays: existence, generalization and alternatives,” Ph.D. thesis, University of London , 1992 ‡ [3] S. Matsufuji, R. Shigemitsu, Y. Tanada, and N. Kuroyanagi, “Construction of complementary arrays,” in Proc. Joint IST Workshop on Mobile Future and Symp. on Trends in Commun. , Bratislava, Slovakia, Oct. 2004, pp. 78–81 § [4] J. Jedwab and M. G. Parker, “Golay complementary array pairs,” Designs, Codes and Cryptography , vol. 44, no. 1-3, p. 209–216, Sep. 2007 ¶ [5] F. Fiedler, J. Jedwab, and M. G. Parker, “A multi-dimensional approach for the construction and enumeration of Golay complementary sequences,” J. Combinatorial Theory (Series A) , vol. 115, no. 5, pp. 753–776, Jul. 2008 � [6] F. Zeng and Z. Zhang, “Two dimensional periodic complementary array sets,” in Proc. IEEE Int. Conf. on Wireless Commun., Netw. and Mobile Comput. , Chengdu, China, Sep. 2010, pp. 1–4 Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 4 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Introduction Main Contribution Contributions: A mapping from generalized Boolean functions to 2-D arrays. A direct construction of 2-D GCAPs based on generalized Boolean functions. A construction of 2-D Golay complementary array mates from Boolean functions. Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 5 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Definition A q -ary array c of size L 1 × L 2 can be expressed as c = ( c g,i ) 0 ≤ g < L 1 , 0 ≤ i < L 2 . The corresponding q -PSK modulated array C is C = ( C g,i ) = ( ξ c g,i ) 0 ≤ g < L 1 , 0 ≤ i < L 2 , where ξ = e 2 π √− 1 /q ; c g,i ∈ { 0 , 1 , · · · , q − 1 } = Z q ; q is an even number. Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 6 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions 2-D Aperiodic Correlation Function Definition (2-D aperiodic cross-correlation function) We define the 2-D aperiodic cross-correlation function of arrays C and D at shift ( u 1 , u 2 ) as ρ ( C , D ; u 1 , u 2 ) = L 1 − 1 − u 1 L 2 − 1 − u 2 ξ d g + u 1 ,i + u 2 − c g,i , 0 ≤ u 1 < L 1 , 0 ≤ u 2 < L 2 ; � � g =0 i =0 L 1 − 1 − u 1 L 2 − 1+ u 2 ξ d g + u 1 ,i − c g,i − u 2 , 0 < u 1 < L 1 , − L 2 < u 2 < 0; � � g =0 i =0 L 1 − 1+ u 1 L 2 − 1+ u 2 ξ d g,i − c g − u 1 ,i − u 2 , − L 1 < u 1 ≤ 0 , − L 2 < u 2 ≤ 0; � � g =0 i =0 L 1 − 1+ u 1 L 2 − 1 − u 2 ξ d g,i + u 2 − c g − u 1 ,i , − L 1 < u 1 < 0 , 0 < u 2 < L 2 . � � g =0 i =0 Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 7 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Golay Complementary Array Pair When C = D , ρ ( C , C ; u 1 , u 2 ) is called the 2-D aperiodic autocorrelation function of C and denoted by ρ ( C ; u 1 , u 2 ) . Definition (Golay Complementary Array Pair) A pair of arrays C and D of size L 1 × L 2 is called a GCAP , if � 2 L 1 L 2 , ( u 1 , u 2 ) = (0 , 0) ρ ( C ; u 1 , u 2 ) + ρ ( D ; u 1 , u 2 ) = 0 , ( u 1 , u 2 ) � = (0 , 0) . C = ( ξ c g,i ) , D = ( ξ d g,i ) where c = ( c g,i ) and d = ( d g,i ) over Z q for 0 ≤ g < L 1 , 0 ≤ i < L 2 . We call ( c , d ) a q -ary GCAP . Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 8 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Golay Complementary Array Mate Definition (Golay Complementary Array Mate) For two GCAPs ( A , B ) and ( C , D ) , they are called the Golay complementary array mate of each other if ρ ( A , C ; u 1 , u 2 ) + ρ ( B , D ; u 1 , u 2 ) = 0 for all ( u 1 , u 2 ) . Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 9 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Generalized Boolean Functions A 2-D generalized Boolean function f : ( y 1 , y 2 , . . . , y n , x 1 , x 2 , . . . , x m ) ∈ Z n + m 2 mapping − → f ( y 1 , y 2 , . . . , y n , x 1 , x 2 , . . . , x m ) ∈ Z q where x i , y g ∈ { 0 , 1 } for i = 1 , 2 , . . . , m and g = 1 , 2 , . . . , n . The monomial of degree r is a product of r variables. For example, x 1 x 2 y 1 y 2 is a monomial of degree 4. Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 10 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Generalized Boolean Functions Let the associated array of size 2 n × 2 m f 0 , 0 f 0 , 1 · · · f 0 , 2 m − 1 · · · f 1 , 0 f 1 , 1 f 1 , 2 m − 1 f = . . . ... . . . . . . · · · f 2 n − 1 , 0 f 2 n − 1 , 1 f 2 n − 1 , 2 m − 1 where f g,i = f (( g 1 , g 2 , · · · , g n ) , ( i 1 , i 2 , · · · , i m )) ; ( g 1 , g 2 , · · · , g n ) is the binary representation of the integer g = � n h =1 g h 2 h − 1 ; ( i 1 , i 2 , · · · , i m ) is the binary representation of the integer i = � m j =1 i j 2 j − 1 . Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 11 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions Generalized Boolean Functions Example 1 Let q = 4 , n = 2 , and m = 3 . The generalized Boolean function is f = 3 x 1 + 2 x 2 x 3 + 2 y 1 + y 2 . The associated array is 0 3 0 3 0 3 2 1 2 1 2 1 2 1 0 3 f = . 1 0 1 0 1 0 3 2 3 2 3 2 3 0 1 0 2 2 × 2 3 Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 12 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions A Construction of 2-D GCAPs Theorem 1 For a q -ary array c , let π 1 be a permutation of { 1 , 2 , · · · , m } and π 2 be a permutation of { 1 , 2 , · · · , n } . Let the generalized Boolean function m − 1 n − 1 f = q � � x π 1 ( i ) x π 1 ( i +1) + y π 2 ( g ) y π 2 ( g +1) 2 i =1 g =1 (1) m n + q � � 2 x π 1 ( m ) y π 2 (1) + p i x i + λ g y g + p 0 i =1 g =1 where p i , λ g ∈ Z q . Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 13 / 22
Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions A Construction of 2-D GCAPs Theorem 1 (Cont’d) The array pair f , f + q � � ( c , d ) = 2 x π 1 (1) (2) is a q -ary GCAP of size 2 n × 2 m . Cheng-Yu Pai (ES, NCKU) ISIT 2020 June 21-26, 2020 14 / 22
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