Introduction Bounds Projective Constructions An Affine Construction References Higher Dimensional Optical Orthogonal Codes Finite Geometries 2017 5 th Irsee Conference Tim Alderson 1 University of New Brunswick September 12, 2017. 1 Supported by the NSERC of Canada Discovery Grants Program. 1 / 28
Introduction Bounds Projective Constructions An Affine Construction References Table of contents Introduction Bounds Projective Constructions An Affine Construction 2 / 28
Introduction Bounds Projective Constructions An Affine Construction References In Optical code-division multiple access (OCDMA) applications, the number of codewords in an OOC corresponds to possible number of asynchronous users able to transmit information efficiently and reliably. 1D-OOCs suffer from small cardinality (need long codewords or relaxed correlations). 3D-OOCs or space/wavelength/time OOCs encode the data bits in spatial, wavelength and time domains, overcoming the 1D-OOC shortcomings. 3 / 28
Introduction Bounds Projective Constructions An Affine Construction References 3D OOCs We denote by (Λ × S × T, w, λ a , λ c ) a 3D-OOC with constant weight w , Λ wavelengths, space spreading length S , and time-spreading length T (hence, each codeword may be considered as an Λ × S × T binary array) subject to the following properties. • (auto-correlation property) for any codeword A = ( a i,j,k ) and for any integer 1 ≤ t ≤ T − 1 , we have S − 1 Λ − 1 T − 1 � � � a i,j,k a i,j,k + t ≤ λ a , i =0 j =0 k =1 • (cross-correlation property) for any two distinct codewords A = ( a i,j,k ) , B = ( b i,j,k ) and for any integer 0 ≤ t ≤ T − 1 , S − 1 Λ − 1 T − 1 � � � we have a i,j,k b i,j,k + t ≤ λ c , i =0 j =0 k =0 where each subscript is reduced modulo T . 4 / 28
Introduction Bounds Projective Constructions An Affine Construction References Example t 3 t 3 t 2 t 2 t 1 t 1 λ 1 λ 1 λ 2 λ 2 λ 3 λ 3 s 1 s 1 s 2 s 2 s 3 s 3 Figure: Autocorrelation λ a = 1 Figure: Autocorrelation zero! Codes with λ a = 0 are called ideal codes . 5 / 28
Introduction Bounds Projective Constructions An Affine Construction References Bounds t 3 t 2 t 1 λ 1 λ 2 λ 3 s 1 s 1 s 2 s 3 A codeword from an ideal 3-D OOC, black cubes indicate 1 , white indicate 0 . (b) Each of the Λ S space/wavelength sections correspond to an element from an alphabet of size T + 1 . 6 / 28
Introduction Bounds Projective Constructions An Affine Construction References Bounds Let Φ( C ) denote the theoretical upper bound on the capacity of C . After adapting the Johnson Bound for non-binary alphabets we obtain the following bound for ideal 3-D OOCs. Theorem [Johnson Bound for Ideal 3D OOC] Let C be an (Λ × S × T, w, 0 , λ ) -OOC, then Φ( C ) ≤ J (Λ × S × T, w, 0 , λ c ) � Λ S � T (Λ S − 1) � � T (Λ S − λ ) �� � = · · · · · · w w − 1 w − λ Note thatif C is an ideal 3D OOC of maximal weight ( w = Λ S ) then Φ( C ) ≤ T λ . Codes meeting the bound will be said to be J-optimal . 7 / 28
Introduction Bounds Projective Constructions An Affine Construction References Bounds One way to achieve λ a = 0 is to select codes with at most one pulse per spatial plane. Such codes are referred to as at most one pulse per plane (AMOPP) codes. AMOPP codes of maximal weight S have a single pulse per spatial plane, and are referred to as SPP codes. 8 / 28
Introduction Bounds Projective Constructions An Affine Construction References Bounds Using similar methods as above we are able to establish that for fixed dimensions, weight, and correlation Φ( SPP ) ≤ Λ λ T λ − 1 ≤ Φ( AMOPP ) � 1 � Λ ST � Λ T ( S − 1) � � Λ T ( S − λ ) ��� ≤ · · · T w w − 1 w − λ ≤ Φ( Ideal ) � Λ S � T (Λ S − 1) � � T (Λ S − λ ) �� � ≤ · · · · · · w w − 1 w − λ 9 / 28
Introduction Bounds Projective Constructions An Affine Construction References Known families of optimal ideal 3D OOC, λ c = 1 . p a prime, q a prime power, θ ( k, q ) = q k +1 − 1 q − 1 Conditions Type Ref. w = S ≤ p for all p dividing Λ T SPP Kim,Yu,Park, (2000) w = S = Λ = T = p SPP Li, Fan, Shum (2012) w = S = 4 ≤ Λ = q , T ≥ 2 SPP Li, Fan, Shum (2012) w = S = q + 1 , Λ = q > 3 , T = p > q SPP Li, Fan, Shum (2012) w = S = 3 Λ ≡ T mod 2 SPP Shum (2015) w = 3 , Λ T ( S − 1) even, AMOPP Shum(2015) Λ T ( S − 1) S ≡ 0 mod 3 , and S ≡ 0 , 1 mod 4 if T ≡ 2 mod 4 and Λ is odd. 10 / 28
Introduction Bounds Projective Constructions An Affine Construction References Projective Spaces: Notation • PG ( k, q ) : The finite projective geometry of dimension k and order q . • The number of points of PG ( k, q ) : θ ( k, q ) = θ ( k ) = q k +1 − 1 . q − 1 • Number of lines on PG ( k, q ) : L ( k ) • The number of d -flats in PG ( k, q ) : � k + 1 = ( q k +1 − 1)( q k +1 − q ) · · · ( q k +1 − q d ) � ( q d +1 − 1)( q d +1 − q ) · · · ( q d +1 − q d ) . d + 1 q 11 / 28
Introduction Bounds Projective Constructions An Affine Construction References Singer representation A Singer group is a cyclic group acting sharply transitively on the points of PG ( k, q ) . A generator is a Singer cycle . Let β be a primitive element of GF ( q k +1 ) . Then the powers of β : β 0 , β 1 , β 2 , . . . , β q k + q k − 1 + ··· + q 2 + q (= θ ( k,q ) − 1) represent the projective points of Σ = PG ( k, q ) . Denote by φ the Singer cycle of Σ defined by β i �→ β i +1 . 12 / 28
Introduction Bounds Projective Constructions An Affine Construction References Codewords from Orbits Let n = θ ( k ) = Λ · S · T where G is the Singer group of Σ = PG ( k, q ) . Since G is cyclic there exists a unique subgroup H of order T ( H is the subgroup with generator φ Λ S ). Definition (Projective Incidence Array) Let Λ , S, T be positive integers such that θ ( k, q ) = Λ · S · T . For an arbitrary pointset A in Σ = PG ( k, q ) we define the Λ × S × T incidence array A = ( a i,j,k ) , 0 ≤ i ≤ Λ − 1 , 0 ≤ j ≤ S − 1 , 0 ≤ k ≤ T − 1 where a i,j,k = 1 if and only if the point corresponding to β i + j · Λ+ k · S Λ is in A . Note that a cyclic shift of the temporal planes of A is the incidence array corresponding to σ ( A ) . 13 / 28
Introduction Bounds Projective Constructions An Affine Construction References β 9 induces a cyclic shift of the temporal planes. t 0 t 1 t 2 λ 0 λ 0 λ 0 β 6 β 15 β 24 β 7 β 16 β 25 λ 1 λ 1 λ 1 β 3 β 8 β 12 β 17 β 21 β 26 β 4 β 13 β 22 λ 2 λ 2 λ 2 β 5 β 14 β 23 β 0 β 9 β 18 β 1 β 10 β 19 β 2 β 11 β 20 t 2 t 1 t 0 λ 0 β 6 β 7 λ 1 β 3 β 8 β 4 λ 2 β 5 β 0 β 1 β 2 14 / 28
Introduction Bounds Projective Constructions An Affine Construction References If A is a pointset of Σ , consider its orbit Orb H ( A ) under the group H generated by φ Λ S . n The set A has full H -orbit if | Orb H ( A ) | = T = Λ S and short H -orbit otherwise. If A has full H -orbit then a representative member of the orbit and corresponding 3-D codeword is chosen. The collection of all such codewords gives rise to a (Λ × S × T, w, λ a , λ c ) -3D-OOC, where | φ Λ S · i ( A ) ∩ φ Λ S · j ( A ) | � � λ a = max (1) 0 ≤ i<j ≤ T − 1 and | φ Λ S · i ( A ) ∩ φ Λ S · j ( A ′ ) | � � λ c = max (2) 0 ≤ i,j ≤ T − 1 ranging over all A , A ′ with full H -orbit. 15 / 28
Introduction Bounds Projective Constructions An Affine Construction References A handy Theorem Theorem ( Rao (1969), Drudge (2002) ) In Σ = PG ( k, q ) , there exists a short G -orbit of d -flats if and only if gcd ( k + 1 , d + 1) � = 1 . In the case that d + 1 divides k + 1 there is a short orbit S which partitions the points of Σ (i.e. constitutes a d -spread of Σ ). There is precisely one such orbit, and the θ ( k ) θ ( d ) � . G -stabilizer of any Π ∈ S is Stab G (Π) = � φ 16 / 28
Introduction Bounds Projective Constructions An Affine Construction References Codes from projective lines, λ c = 1 In PG ( k, q ) , k odd, let S be the line spread determined by G where say Stab G ( ℓ ) = H for ℓ ∈ S (so | H | = q + 1 ). It follows that any pointset meeting each line of the spread in at most one point will be of full H -orbit, and moreover, that members of the orbit will be mutually disjoint. (Consequently, if Λ S = θ ( k,q ) q +1 , then the corresponding Λ × S × ( q + 1) incidence array will satisfies λ a = 0 ). 17 / 28
Introduction Bounds Projective Constructions An Affine Construction References Clearly, each line ℓ / ∈ S meets each spread line in at most one point. For each full H -orbit of lines, select a representative member and corresponding Λ × S × ( q + 1) incidence array (3D-codeword). The collection of all such codewords comprises a (Λ × S × ( q + 1) , q + 1 , 0 , λ c ) -3DOOC C . As two lines intersect in at most one point we have λ c = 1 . 18 / 28
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