Some Combinatorial Packing Problems Arising from Multiple Access
Some Combinatorial Packing Problems Arising from Multiple. Access Collision Communications Kenneth Shum Joint work with Pingzhi Fan, Hung-Lin Fu , Xiyang Li, Yuan-Hsun Lo, Wing Shing Wong, and Yijin Zhang
Outline Binary Sequences, Hamming crosscorrelation Multi-dimensional optical orthogonal codes Conflict-avoiding codes Additive combinatorics Mar 2015 Combinatorial design theory kshum 2
Multiple-access collision channel without feedback Hello ! message Channel coding (Reed-Solomon) Packet 1 Any two packets can recover the original message Packet 2 Packet 3 Packet 4 x x ion collis Other user collision collis ion Scheduler Other user x Receiver Hello ! Mar 2015 kshum 3
Binary sequences and Hamming correlation function • Periodic binary sequences u(t) and v(t) with period T. – t = 0, 1, 2, 3, . . . • The Hamming cross-correlation function of u(t) and v(t) is defined as – is the relative delay offset between the two sequences. – denotes the modulo-T addition. • The Hamming cross-correlation counts the number of overlapping ones in u(t) and v(t ). • When u = v, Huu( ) is called the Hamming auto-correlation of u(t). • Huu(0) is counts the number of 1 in a period of u(t), and is called the Hamming weight of u(t). Mar 2015 kshum 4
Example • • Period T = 6 u(t) : 111000 v(t) : 101010 Both u(t) and v(t) have Hamming weight 3. Mar 2015 Huv(0) = 2 111000 101010 Huv(1) = 1 111000 010101 kshum 5
Definitions • Optical orthogonal code OOC(T, , a, c) – – Period T Weight Hamming auto-correlation a Hamming cross-correlation c • Conflict-avoiding code (Tsybakov and Rubinov (02)) CAC(T, ) = OOC(T, , , 1) – – Period T Weight Hamming auto-correlation Hamming cross-correlation c No requirement on Hamming auto-correlation. Mar 2015 kshum 6
Parameters • Number of codewords = m – Total number of potential users – Each user is statically assigned a unique codeword • Sequence period = T – maximal delay experience by an active user • Hamming weight = – Maximal number of simultaneously active users • Objective: Given T and , maximize m Mar 2015 kshum 7
Maximal number of codewords • Let M(T, ) be largest number of codewords in a conflictavoiding codes of length T and weight . • A conflict-avoiding codes of length T and weight containing M(T, ) codewords is said to be optimal. • Levenshtein (07) for T = 4 k + 2 for odd T, T Mar 2015 kshum 8
CAC of even length and weight 3 • For T = 4 k, • Jimbo, Mishima, Janiszewski, Teymorian and Tonchev (07) • Mishima, Fu and Uruno (09) • Fu, Lin and Mishima (10) Mar 2015 kshum 9
CAC of weight > 3 • Some constructions of optimal CAC of weight 4 and 5 – Momihara, Müller, Satoh and Jimbo (07) • CAC in general – S and Wong For 3, Mar 2015 kshum 10
Terminology • A binary sequence can be represented by a characteristic set. – Sequence: 0 1 1 0 0 {1, 2, 5} indices 0 1 2 3 4 5 6 7 • The set of differences contains the separations between the ones in a sequence – (A) : = {x – y mod T: x, y A, x y} – For example ({1, 2, 5}) = {1, 3, 4, 5, 7} 011001000110100 Mar 2015 11
CAC as a packing problem (A) = {x – y mod T: x, y A, x y} • The characteristic sets of CAC is a collection of subsets of {0, 1, …, T– 1}, say A 1, A 2, …, AM , such that – Each of them has size . – (Ai) (Ak) = for i k. • Example: T=15, – – Mar 2015 distinct 111000000 {0, 1, 2}, ({0, 1, 2}) = {1, 2, 13, 14} 10010010000 {0, 3, 6}, ({0, 3, 6}) = {3, 6, 9, 12} 10001000000 {0, 4, 8}, ({0, 4, 8}) = {4, 7, 8, 11} 1000010000 {0, 5, 10}, ({0, 5, 10}) = {5, 10} 12
Kneser’s theorem • Due to Martin Kneser (1953) – An extension of Cauchy-Davenport theorem. • Let – G be a finite abelian group, – A and B be non-empty subsets of G, – A+B : = {a+b: a A, b B} be the sumset of A and B. • If |A| + |B| |G|, then there is a subgroup H of G such that |A+B| |A|+|B|– |H|. – H is the stablizer of A+B. H : = {g G: g+(A+B) = A+B}. • Apply Kneser’s theorem with G = ZT and B = – A : = {– a: a A}. • If 2|A| T, then there is a divisor h of T such that (A) 2|A|– 1–h. – Mar 2015 (A) 2|A|– 2 if the stablizer H is the trivial subgroup. kshum 13
Upper bound on the size of a CAC • For any CAC of period T and weight 3, • It can be shown that equality holds by explicitly constructing a sequence of CAC attaining the bound. • The number of codewords scales linearly with the period. Mar 2015 kshum 14
Outline Binary Sequences, Hamming crosscorrelation Multi-dimensional optical orthogonal codes Conflict-avoiding codes Kneser’s theorem Mar 2015 kshum 15
Recall: Optical orthogonal code • A set of binary sequences of period T is an optical orthogonal code, denoted by OOC(T, , a, c), if – All sequences have Hamming weight w. • Huu(0) = , for all sequence u. – All sequences have Hamming auto-correlation less than or equal to a for all 0. • Huu( ) a, for sequence u and 0. – The Hamming cross-correlation between each pair of sequences is less than or equal to c for all . • Huv( ) c, for all pairs of distinct sequences u and v, and delay offset . Mar 2015 kshum 16
2 -D optical-orthogonal code • 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 T satisfying – 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 in C , and for all . Mar 2015 kshum 17
3 -D optical-orthogonal code • 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 T satisfying – 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 . Mar 2015 kshum 18
Applications • Optical Code-Division Multiple Access – Spreading in spatial, frequency and time domains. – Transmit an optical pulse if and only if the corresponding entry in the 3 -D array is 1. • Digital watermarking for video signal – The cross-sections of a 3 -D array in the time axis are associated with a frame of in a video. – Mark a pixel if and only if the corresponding entry in the 3 -D array is 1. Mar 2015 kshum 19
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. Codeword 8 Codeword 7 Codeword 6 Codeword 5 Mar 2015 Codeword 4 wavelength 0 wavelength 1 Codeword 3 Second spatial plane Codeword 2 wavelength 0 wavelength 1 Codeword 1 First spatial plane 20
Terminology A spatial plane is a wavelength/time plane (for a fixed index of spatial channel). • At-most-one-pulse-per-plane code (AMOPPC) a = 0 • – At most one optical pulse in each spatial plane. • Single-pulse-per-plane code (SPPC) – Exactly one optical pulse in each spatial plane. • S a = 0 S= Multiple-pulse-per-plane code (MPPC) – More than one optical pulse in each spatial plane. • At-most-one-pulse-per-time code (AMOPTC) T Single-pulse-per-time code (SPTC) T= – Transmit at most one optical pulse in each time slot. • – Transmit exactly one optical pulse in each time slot. Mar 2015 kshum 21
Existing constructions Ref. S, W, T |C| Type [1] All prime factors of T SW SW 1 T MPPC [1] All prime factors of W and T S S 1 W 2 T SPPC [2] S = T = p, W = p 2 -1, p prime p 2 -1 1 p(p 2 -1) SPTC [2] S = W = p, T = p 2 -1 1 p 2 SPTC [3] S = W = T = p, p prime, 1 r p-2 S r W +1 T SPPC [3] S =4, W=q, T 2, q is a prime power 4 S 2 W 3 T 2 SPPC [3] S = q+1, W = q, T = p, q is a prime power 4, p is a prime q 3 1 W 2 T SPPC [4] S = 3, W is even when T is even 3 1 W 2 T SPPC [4] (S-1)WT even, 3|S(S-1)WT S 0, 1 mod 4 when T 2 mod 4 and W 1 mod 2 3 1 [1] Kim, Yu and Park, 2000. [2] Ortiz-Ubarri, Moreno and Tirkel, 2011. [3] Li, Fan and S, 2012. [4] S, 2015. Mar 2015 kshum AMOPPC 22
Johnson-type bound • For 3 -D OOC in general, [2] • For the class of at-most-one-pulse-per-plane code, Ortiz-Ubarri, Moreno and Tirkel, 2011. S, 2015. Mar 2015 kshum 23
Perfect 3 -D AMOPPC Remove all the floor operators • A 3 -D at-most-one-pulse-per-plane code attaining equality in the above bound on code size is called perfect. Mar 2015 kshum 24
Characterization of perfect 3 -D OOC Theorem 3: The followings are equivalent: • A perfect (S W T, , 0, 1)-AMOPPC. • ( , T; ZT)-GBRGDD of type WS. First spatial plane wavelength 0 wavelength 1 Second spatial plane wavelength 0 wavelength 1 S=W=T= =2 , =1 Mar 2015 0 0 0 0 0 1 0 1 25
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 an n 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 (x 1, x 2, …, xb) and (y 1, y 2, …, yb), the list xi – yi: i = 1, 2, …, b, xi , yi , contains exactly copies of each element of G. • The parameters are certainly not independent. – is a multiple of |G|. – bk = rn. – r(k – 1) = (n – 1) • If we replace by 0, and replace group elements in G by 1, then what we get is the incident matrix of a balanced incomplete block design. • The GBRD is said to be obtained by signing the incidence matrix by G. Example: (4, 3, 6; Z 6)-GBRD 0 0 1 1 1 0 3 3 0 0 0 3 3 0 1 1 0 1 1 2 0 2 4 0 0 1 4 3 0 2 1 0 26
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}. Mar 2015 kshum 27
Example • • 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} } Group 1 1 2 3 Group 2 4 5 Group 3 Type = 1 22 Mar 2015 kshum 28
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 ms by 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)m 2|G| / (k(k – 1)). Mar 2015 kshum 29
Perfect 3 -D OOC of weight 3 • Theorem 4: A perfect (3 W T)-single pulse per plane code exists if and only if – T is odd, or – T is even and W is even. • Theorem 5: For S > 3, a perfect (3 W T)-at most one pulse per plane code exists if and only if the following three conditions hold simultaneously – (S – 1)WT 0 mod 2, – S(S – 1)WT 0 mod 3, and – S 0, 1 mod 4 when T 2 mod 4 and W 1 mod 2. Mar 2015 kshum 30
Outline Binary Sequences, Hamming crosscorrelation Multi-dimensional optical orthogonal codes Conflict-avoiding codes Kneser’s theorem Mar 2015 Generalized Basker Rao group divisible design kshum 31
References • • • Tsybakov and Rubinov, Some constructions of conflict-avoiding codes, Problems of Inf. Trans. , 2002. V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Probems of Inf. Trans. , 2007. M. Jimbo et al. , On conflict-avoiding codes of length n=4 m for three active users, IEEE Trans. Information Theory, 2007. M. Mishima, H. -L. Fu and S. Uruno, Optimal conflict-avoiding codes of length n 0(mod 16) and weight 3, Des. Codes Cryptogr. , 2009. H. -L. Fu, Y. -H. Lin and M. Mishima, Optimal conflict-avoiding codes of even length and weight 3, IEEE Trans. Inf. Theory, 2010. K. W. Shum and W. S. Wong, A tight asymptotic bound on the size of constant-weight conflictavoiding codes, Designs, Codes and Cryptography, 2010. Y. Zhang, K. W. Shum and W. S. Wong, Strongly conflict-avoiding codes, SIAM J. Discrete Math. 2011. H. -L. Fu, Y. -H. Lo and K. W. Shum, Optimal conflict-avoiding codes of odd length and weight three, to appear in Designs, Codes and Cryptography. S. Kim, K. Yu and N. Park, A new family of space/wavelength/time spread three-dimensional optical orthogonal code for OCDMA network, J. lightwave Tech. 2000. J. Ortiz-Ubarri, O. Moreno and A. Tirkel, Three-dimensional periodic optical orthogonal code fo OCDMA systems, IEEE Information Theory Workshop, 2011. X. Li, P. Fan and K. W. Shum, Construction of space/wavelength/time spread optical orthogonal code with large family size, IEEE Comm. Letter, 2012. K. W. Shum, Optimal three-dimensional optical orthogonal codes of weight three, to appear in Designs, Codes and Cryptography. 32
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