Director of Research (if dissertation) or Advisor (if thesis)
Duursma, Iwan M.
Doctoral Committee Chair(s)
Reznick, Bruce
Hajek, Bruce
Committee Member(s)
Duursma, Iwan M.
Schenck, Henry K.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
coding theory
Reed-Muller codes
secret sharing
multiparty computation
combinatorics
multiplicity
Abstract
This is a comprehensive study of multiplicative codes of Reed-Muller type and their applications.
Our codes apply to the elds of cryptography and coding theory, especially to multiparty computa-
tion and secret sharing schemes. We also study the AB method to analyze the minimum distance
of linear codes. The multiplicative codes of Reed-Muller type and the AB method are connected
when we study the distance and dual distance of a code and its square. Generator matrices for our
codes use a combination of blocks, where a block consists of all columns of a given weight. Several
interesting linear codes, which are best known linear codes for a given length and dimension, can
be constructed in this way.
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