47,087 research outputs found
Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions
We introduce the class of multiply constant-weight codes to improve the
reliability of certain physically unclonable function (PUF) response. We extend
classical coding methods to construct multiply constant-weight codes from known
-ary and constant-weight codes. Analogues of Johnson bounds are derived and
are shown to be asymptotically tight to a constant factor under certain
conditions. We also examine the rates of the multiply constant-weight codes and
interestingly, demonstrate that these rates are the same as those of
constant-weight codes of suitable parameters. Asymptotic analysis of our code
constructions is provided
Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes
Multiply constant-weight codes (MCWCs) have been recently studied to improve
the reliability of certain physically unclonable function response. In this
paper, we give combinatorial constructions for MCWCs which yield several new
infinite families of optimal MCWCs. Furthermore, we demonstrate that the
Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights
and distances. Finally, we provide bounds and constructions of two dimensional
MCWCs
Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers
We transfer the concept of linear feed-back shift registers to quantum
circuits. It is shown how to use these quantum linear shift registers for
encoding and decoding cyclic quantum error-correcting codes.Comment: 18 pages, 15 figures, submitted to Proc. R. Soc.
Code algebras, axial algebras and VOAs
Inspired by code vertex operator algebras (VOAs) and their representation
theory, we define code algebras, a new class of commutative non-associative
algebras constructed from binary linear codes. Let be a binary linear code
of length . A basis for the code algebra consists of idempotents
and a vector for each non-constant codeword of . We show that code algebras
are almost always simple and, under mild conditions on their structure
constants, admit an associating bilinear form. We determine the Peirce
decomposition and the fusion law for the idempotents in the basis, and we give
a construction to find additional idempotents, called the -map, which comes
from the code structure. For a general code algebra, we classify the
eigenvalues and eigenvectors of the smallest examples of the -map
construction, and hence show that certain code algebras are axial algebras. We
give some examples, including that for a Hamming code where the code
algebra is an axial algebra and embeds in the code VOA .Comment: 32 pages, including an appendi
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