Matrix Algebras and Semidefinite Programming Techniques for Codes
Dion Gijswijt
TL;DR
This thesis explores advanced semidefinite programming bounds for nonbinary error-correcting and covering codes, utilizing matrix algebra techniques like block-diagonalisation and Terwilliger algebras to improve bounds and computational methods.
Contribution
It introduces novel SDP bounds for codes using matrix algebra methods, including explicit block-diagonalisation of Terwilliger algebras, advancing coding theory bounds.
Findings
New SDP bounds for nonbinary codes and coverings
Explicit block-diagonalisation of Terwilliger algebra
Computational results for affine caps
Abstract
This PhD thesis is concerned with SDP bounds for codes: upper bounds for (non)-binary error correcting codes and lower bounds for (non)-binary covering codes. The methods are based on the method of Schrijver that uses triple distances in stead of pairs as in the classical Delsarte bound. The main topics discussed are: 1) Block-diagonalisation of matrix *-algebras, 2) Terwilliger-algebra of the nonbinary Hamming scheme (including an explicit block-diagonalisation), 3) SDP-bounds for (nonbinary) error-correcting codes and covering codes (including computational results), 4) Discussion on the relation with matrix-cuts, 5) Computational results for Affine caps.
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Taxonomy
TopicsAdvanced Optimization Algorithms Research · VLSI and FPGA Design Techniques · graph theory and CDMA systems