# Combinatorial Designs and Error Correcting Codes

Vijay Bhargava

## Abstract

It is well known that several combinatorial designs have properties that can be used to construct good error correcting codes. In this talk we use the properties of the incidence matrix of (v, k, ?) designs to construct some really elegant error correcting codes. If v is a twin prime or prime power such that v = 3 (mod 4), then (v, k, ?) = (v, (v – 1)/2, (v – 3)/4) or its complement (v, k*, ?*) = (v, (v + 1)/2, (v+1)/4) are called Hadamard Designs. One can then generate a (2v+2, v+1) by a generator matrix of the form G = [I, Sv] where Sv is derived from the incidence matrix of the Hadamard Design. If v = 3 (mod 8), the resulting codes can be shown to be self-dual with weights divisible by four. It can also be shown that these codes are related to the binary images of (v+1, (v+1)/2) extended quaternary quadratic residue codes. Although the material is inherently mathematical in nature, it will be presented in an understandable manner.