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Optimal Three-Dimensional Optical Orthogonal Codes and Related Combinatorial Designs

Optimal Three-Dimensional Optical Orthogonal Codes and Related Combinatorial Designs. Kenneth Shum. Outline. Three-dimensional optical orthogonal codes Existing constructions Bound on number of codewords Related combinatorial designs Group divisible designs

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Optimal Three-Dimensional Optical Orthogonal Codes and Related Combinatorial Designs

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  1. Optimal Three-Dimensional Optical Orthogonal Codes and Related Combinatorial Designs Kenneth Shum

  2. Outline • Three-dimensional optical orthogonal codes • Existing constructions • Bound on number of codewords • Related combinatorial designs • Group divisible designs • Generalized Bhaskar Rao designs.

  3. 1-D optical-orthogonal code • Spreading in time domain. • For two binary sequences x(t)and y(t) of length T, the Hamming correlation function counts the number of overlapping ones after we cyclically shift y(t) by , • A (T,, a, c) 1-dimensional OOC is a set C of binary sequences of length T satisfying • Hxx(0) = , for all x in C, (constant weight) • Hxx()  a, for all x in C and for all nonzero , • Hxy()  c, for distinct x and y in C, and all .

  4. 2-D optical-orthogonal code • Spreading in time and frequency domains. • For two binary 2-dimensional arrays X(w,t) and Y(w,t) of size WT, the Hamming correlation function counts the number of overlapping ones after we cyclically shift Y(w,t) in the second dimension by , • A (WT, , a, c) 2-dimensional OOC is a set C of binary arrays of size WTsatisfying • HXX(0) =  for all X in C, (constant weight) • HXX()  a, for all X in C and for nonzero , • HXY()  c, for distinct X and Y, and all .

  5. 3-D optical-orthogonal code • Spreading in spatial, frequency and time domains. • For two binary 3-dimensional arrays X(s,w,t) and Y(s,w,t) of size SWT, the Hamming correlation function counts the number of overlapping ones after we cyclically shift Y(s,w,t) in the last dimension by , • A (SWT, , a, c) 3-dimensional OOC is a set C of binary arrays of size SWTsatisfying • HXX(0) =  for all X in C, (constant weight) • HXX()  a, for all X in C and for nonzero , • HXY()  c, for distinct X and Y, and all .

  6. A small example • ( 222, 2, 0, 1) 3-D OOC • S=2 spatial channels • W=2 wavelength • T=2 time chips in a period. • Hamming weight  = 2 • a = 0, zero auto-correlation • c = 1, cross-correlation upper bounded by 1. wavelength 0 wavelength 1 First spatial plane Second spatial plane wavelength 0 wavelength 1 Codeword 7 Codeword 8 Codeword 1 Codeword 2 Codeword 3 Codeword 4 Codeword 5 Codeword 6

  7. Different classes of 3-D OOC • A spatial plane is a wavelength/time plane (for a fixed index of spatial channel). • At-most-one-pulse-per-plane code (AMOPPC) • At most one optical pulse in each spatial plane. • Single-pulse-per-plane code (SPPC) • Exactly one optical pulse in each spatial plane. • Multiple-pulse-per-plane code (MPPC) • More than one optical pulse in each spatial plane. • At-most-one-pulse-per-time code (AMOPTC) • Transmit at most one optical pulse in each time slot. • Single-pulse-per-time code (SPTC) • Transmit exactly one optical pulse in each time slot. a = 0 S a = 0 S=

  8. Existing constructions [1] Kim, Yu and Park, 2000. [2] Ortiz-Ubarri, Moreno and Tirkel, 2011. [3] Li, Fan and S, 2012. [4] S, 2013.

  9. Johnson-type bound • For 3-D OOC in general, [2] • For the class of at-most-one-pulse-per-plane code, [4] [2] Ortiz-Ubarri, Moreno and Tirkel, 2011. [4] S, 2013.

  10. Perfect 3-D AMOPPC • A 3-D at-most-one-pulse-per-plane code satisfying the second bound on code size is called perfect. Remove all the floor operators

  11. Group divisible designs • Let v be a positive integer, K be a set of positive integers, and  be a positive integer. • A group divisible design GDD(K;v) of order v is and block sizes from K is a triple (V,G,B) where • V is a set of size v, called points. • G is a partition of point set V, called groups, • B is a collection of subsets in V, called blocks, s.t. • each block in B has size in K. • each block intersects every group in G in at most one point. • any pair of points from two distinct groups is contained in exactly  blocks of B. • The type of a GDD is the multi-set of group sizes. • i.e., the multi-set {|H|: H  G}.

  12. Example If all groups have size 1and all blocks have the same size, then GDDreduces to BIBD. • v = 5, K= {2,3},  = 1. • V = {1,2,3,4,5}. • G = { {1}, {2,3}, {4,5} } • B = { {1,2,4}, {1,3,5}, {2,5}, {3,4} } 1 Group 1 2 3 Group 2 Type = 1 , 22 4 5 Group 3

  13. Generalized Bhaskar Rao design • Let G be a finite abelian group, and  be a special symbol not in G. • A generalized Bhaskar Rao design, (n, k, ; G)-GBRD, is ann  b array with entries in G  {}, such that • Each row has exactly r entries in G. • Each column contains exactly k entries in G. • Each pair of distinct rows (x1, x2, …, xb) and (y1, y2, …, yb), the list xi – yi: i = 1,2,…,b, xi  , yi  , contains exactly  copies of each element of G. • The parameters satisfy: •  is a multiple of |G|. • bk= rn. • r(k –1) = (n – 1) • If we replace  by 0, andreplace group elements in G by 1,then what we get is the incident matrixof a balanced incomplete block design. • The GBRD is said to be obtained bysigning the incidence matrix by G. Example: (4, 3, 6; Z6)-GBRD

  14. Generalized Bhaskar Rao group divisible design • If we start from a group divisible design, and sign the corresponding incidence matrix by the element in a finite group G, then the resulting matrix is called a generalized Bhaskar Rao group divisible design (GBRGDD) • If we sign a GDD(K; ms) of type msby abelian group G, the resulting GBRGDD is denoted by (K, ; G)-GBRGDD of type ms. • if K is a singleton {k}, we write (k, ; G)-GBRGDD of type ms. • Number of rows = ms. • Number of columns = s(s – 1)m2|G| / (k(k – 1)).

  15. Characterization of perfect 3-D OOC NEW Theorem: The followings are equivalent: • A perfect (SWT,,0,1)-AMOPPC. • (,T; ZT)-GBRGDD of type WS. wavelength 0 wavelength 1 First spatial plane Second spatial plane wavelength 0 wavelength 1 S=W=T==2 , =1

  16. Existing result on existence of GDD • Theorem [5]: Let m and s be positive integers. A necessary and sufficient condition for the existence of GDD(4; ms) of type ms is that the design is not GDD1(4; 8) of type 24, and not GDD1(4,24) of type 64, and such that  (4, ;{e})-GBRGDD of type ms  (sm1, 4,0,1)-AMOPPC with T=1 [5] Brouwer, Schrijver, and Hanani, 1977.

  17. Existing result on existence of GBRD • For an integer g with prime factorization p1p2p3…pd, an elementary abelian group, EA(n), of order g is the direct product of Z/piZ, for i =1, 2 ,…, d. • Theorem [6]: If g divides  , then a (4,4,; EA(g))-GBRD exists unless • g   2 mod 4 when g is even, • g =  = 3 when g is odd.  (4, ;EA(g))-GBRGDD of type 1s  (s1g, 4,0, )-AMOPPC with W=1 [6] Ge, Greig and Seberry, 2003.

  18. Product construction Theorem [6]: Given a perfect (SW1T1, k, 0,1)-AMOPPC and a perfect (kW2T2, , 0,1)-AMOPPC, we can construct a perfect (S W1W2 T1T2, , 0,1)-AMOPPC. Proof by a product construction of GBRGDD in [6], translated to the setting ofAMOPPC by the one-to-one correspondence between GBRGDD over ZTAMOPPC. [6] Ge, Greig and Seberry, 2003.

  19. New constructions of perfect 3-D OOC 19

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