35 research outputs found
On the Kernel of -Linear Hadamard Codes
The -additive codes are subgroups of ,
and can be seen as a generalization of linear codes over and
. A -linear Hadamard code is a binary Hadamard
code which is the Gray map image of a -additive code. It is
known that the dimension of the kernel can be used to give a complete
classification of the -linear Hadamard codes. In this paper, the
kernel of -linear Hadamard codes and its dimension are
established for . Moreover, we prove that this invariant only provides a
complete classification for some values of and . The exact amount of
nonequivalent such codes are given up to for any , by using
also the rank and, in some cases, further computations
Codes from adjacency matrices of uniform subset graphs
Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Î(n,k,r)Î(n,k,r) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from Î(n,3,r)Î(n,3,r) for pâ„5pâ„5 , thus extending earlier results for p=2,3p=2,3
Symmetric-key Corruption Detection : When XOR-MACs Meet Combinatorial Group Testing
We study a class of MACs, which we call corruption detectable MAC, that is able to not only check the integrity of the whole message, but also detect a part of the message that is corrupted.
It can be seen as an application of the classical Combinatorial Group Testing (CGT) to message authentication.
However, previous work on this application has inherent limitation in communication.
We present a novel approach to combine CGT and a class of linear MACs (XOR-MAC) that enables to break this limit. Our proposal, XOR-GTM, has a significantly smaller communication cost than any of the previous ones, keeping the same corruption detection capability. Our numerical examples for storage application show a reduction of communication by a factor of around 15 to 70 compared with previous schemes.
XOR-GTM is parallelizable and is as efficient as standard MACs.
We prove that XOR-GTM is provably secure under the standard pseudorandomness assumptions