CODES AND DESIGNS Tuvi Etzion Computer Science Department

CODES AND DESIGNS Tuvi Etzion Computer Science Department Mathematics of Communications: Sequences, Codes, and Designs Banff, Canada, 26 January 2015 CODES AND DESIGNS

CODES AND DESIGNS Fano plane (projective plane of order 2) generator matrix Hamming code of length 7 CODES AND DESIGNS

CODES Error-Correcting Codes Convolutional Codes Quantum Codes Space Time Codes Authentication Codes Frameproof Codes Network Codes Distributed Storage Codes Reed Solomon Codes Reed Muller Codes Permutation Codes Constrained Codes Covering Codes Subspace Codes Perfect Codes Polar Codes COD ES AND DESIG NS 3

DESIGNS Combinatirial Designs Block Designs Group Divisible Designs Howell Designs Resolvable Designs Covering Designs Steiner Systems Large Sets Latin Squares Orthogonal Arrays Difference Sets Hadamard Matrices Projective Planes Generalized Polygons Transversal Designs Room Squares CODE S AND DESI GNS 4

CODES AND DESIGNS 1. Designs embedded in codes 2. Codes “equivalent” to designs 3. Designs formed from codes 4. Codes formed from designs 5. Combination of codes + designs Steiner systems in perfect codes Steiner system is a constant weight code MDS code is an orthogonal array 1 -parallelisms from Preparata code q-analog designs from rank-metric codes Plotkin’s bound and Hadamard matrices network codes from orthogonal Latin squares Large sets from Steiner systems, orthogonal arrays and Hamming code CODES AND DESIGNS

OUTLINE Basic Concepts MDS Codes Constant Weight Codes Perfect Codes and Designs Projective Spaces Applications CODES AND DESIGNS

BASIC CONCEPTS - DESIGNS CODES AND DESIGNS

BASIC CONCEPTS - DESIGNS CODES AND DESIGNS

BASIC CONCEPTS - DESIGNS CODES AND DESIGNS

BASIC CONCEPTS - CODES AND DESIGNS

BASIC CONCEPTS - CODES AND DESIGNS

BASIC CONCEPTS - CODES AND DESIGNS

MDS CODES Singleton 1964 Golomb, Posner 1964 CODES AND DESIGNS

MDS CODES MDS conjecture Partial proof (not all parameters) for the MDS conjecture Ball 2012 Ball, de Beule 2012 Constructions for orthogonal arrays with index unity for a given length and each strength Blanchard 1995 CODES AND DESIGNS

CONSTANT WEIGHT CODES AND DESIGNS

CONSTANT WEIGHT CODES AND DESIGNS

CONSTANT WEIGHT CODES Keevash 2014 CODES AND DESIGNS

CONSTANT WEIGHT CODES Hanani 1960 Hanani 1979 Blanchard 1995 Wilson 1972 CODES AND DESIGNS

CONSTANT WEIGHT CODES AND DESIGNS

CONSTANT WEIGHT CODES optical orthogonal codes also called cyclically permutable codes Chung, Salehi, Wei 1989 Chung, Kumar 1990 A, Gyorfi, Massey 1992 Bitan, E. 1995 Kumar, Moreno, Zhang, Zinoviev 1993 Wilson 1972 Buratti 1995, 1998, 2002 CODES AND DESIGNS

CONSTANT WEIGHT CODES There are several methods which produces long van Pul, E. 1989 and large constant weight codes (or large Brouwer, general codes) from union of direct products of Shearer, Sloane, disjoint shorter constant weight codes. Smith 1990 Baranyai 1975 CODES AND DESIGNS

CONSTANT WEIGHT CODES Lu 1983, 1984 Teirlinck 1991 (last 6 cases) E. , Hartman 1991 The construction combines disjoint orthogonal arrays, disjoint quadruple Steiner systems, and boolean quadruple Steiner systems which are related to the codewords of weight four in the Hamming code and its cosets. CODES AND DESIGNS

PERFECT CODES AND DESIGNS We will consider only metrics in which all balls with the same radius have the same size. sphere packing bound CODES AND DESIGNS

PERFECT CODES AND DESIGNS

PERFECT CODES AND DESIGNS Applying on the known binary perfect codes (Hamming, Golay) CODES AND DESIGNS

PERFECT CODES AND DESIGNS E. 1996 E. , Schwartz 2004 CODES AND DESIGNS

PERFECT CODES AND DESIGNS Delsarte 1973 Ahlswede, Aydinian, Khachatrian 2001 Chihara 1987 Martin, Zhu 1995 CODES AND DESIGNS

PERFECT CODES AND DESIGNS Ahlswede, Aydinian, Khachatrian 2001 An orthogonal array with index unity (and similarly an MDS code) is a diameter perfect code in the Hamming graph. CODES AND DESIGNS

PROJECTIVE SPACES Fano plane (projective plane of order 2) CODES AND DESIGNS

PROJECTIVE SPACES CODES AND DESIGNS

PROJECTIVE SPACES CODES AND DESIGNS

PROJECTIVE SPACES Koetter, Kschischang 2008 CODES AND DESIGNS

PROJECTIVE SPACES Braun, E. , Östergård, Vardy, Wassermann 2013 15 cyclic shift Frobenius map normalizer of Singer subgroup automoprphism CODES AND DESIGNS

PROJECTIVE SPACES Baker 1976 Baker, van Lint, Wilson 1983 Beutelspacher 1974 E. , Vardy 2012 Sarmiento 2002 Beutelspacher 1990 E. 2014 CODES AND DESIGNS

PROJECTIVE SPACES CODES AND DESIGNS

PROJECTIVE SPACES E. , Silberstein 2013 CODES AND DESIGNS

APPLICATIONS MULTICAS T CODES AND DESIGNS

APPLICATIONS x, y ROUTIN G x MIN-CUT/MAX-FLOW y y x y y y x, y y CODES AND DESIGNS

APPLICATIONS NETWORK CODING x, y x y y x x x+y x, y y x+y x, y CODES AND DESIGNS

APPLICATIONS NETWORK CODING x, y x y y x x x+y x, y y x+y x, y CODES AND DESIGNS

APPLICATIONS A, B Latin squares of order n x x x, y y x y orthogonal A, B x y y Riis, Ahlswede 2006 x A(x, y) y B(x, y) A(x, y) x, y B(x, y) x, y x, y CODES AND DESIGNS

APPLICATIONS WOM Codes (Write Once Memories) information first generation second generation 01 10 11 Rivest, Shamir 1982 Reused today: coding for flash memories CODES AND DESIGNS

APPLICATIONS second generation first generation 1 00 100000 1 2 0 4 2 4 3 5 6 7 CODES AND DESIGNS

APPLICATIONS second third generation 1 01000 10 11 2 4 3 5 6 7 CODES AND DESIGNS

APPLICATIONS fourth third generation 1 1100 11 2 3 1 4 3 5 6 7 CODES AND DESIGNS

APPLICATIONS fourth third generation 1 111 01 00 1 2 6 1 4 3 5 6 7 CODES AND DESIGNS

THANK YOU CODES AND DESIGNS