# Accession Number:

## AD0602991

# Title:

## RESEARCH IN NONLINEAR CODES.

# Descriptive Note:

## Final rept.,

# Corporate Author:

## SYLVANIA ELECTRIC PRODUCTS INC WALTHAM MASS

# Personal Author(s):

# Report Date:

## 1964-03-01

# Pagination or Media Count:

## 72.0

# Abstract:

The first part of this report deals with difference sets, subsets of a group with the property that any two translates are a fixed Hamming distance apart. When the group is cyclic, these correspond to sequences whose periodic autocorrelation function is two-valued. The groups considered are those which admit a large number of characters. Several nonexistence theorems are proved. These assert that if the parameters v and n of a difference set have factors in common, there are bounds on the order of the characters of the group. When v 4n, the difference set gives rise to an Hadamard matrix. The study of Barker sequences, binary sequences with non-periodic autocorrelation function 1 in absolute value except at zero showed that there were no Barker sequences of odd length 13, and any of even length greater than 2 arose from a difference set with v 4n, in a cyclic group or order v. The only ones known of even length are of lengths 2 or 4. The last section applies the theorems proved in the first section to demonstrate the nonexistence of Barker sequences of length v 4 . 39 to the second power 6084 as well as other difference sets with v 4n. The other two sections deal with multipliers of difference sets and the existence of difference sets in elementary abelian groups. Author