35 research outputs found

    On the Kernel of Z2s\mathbb{Z}_{2^s}-Linear Hadamard Codes

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    The Z2s\mathbb{Z}_{2^s}-additive codes are subgroups of Z2sn\mathbb{Z}^n_{2^s}, and can be seen as a generalization of linear codes over Z2\mathbb{Z}_2 and Z4\mathbb{Z}_4. A Z2s\mathbb{Z}_{2^s}-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s\mathbb{Z}_{2^s}-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z4\mathbb{Z}_4-linear Hadamard codes. In this paper, the kernel of Z2s\mathbb{Z}_{2^s}-linear Hadamard codes and its dimension are established for s>2s > 2. Moreover, we prove that this invariant only provides a complete classification for some values of tt and ss. The exact amount of nonequivalent such codes are given up to t=11t=11 for any s≄2s\geq 2, by using also the rank and, in some cases, further computations

    Codes from adjacency matrices of uniform subset graphs

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    Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Γ(n,k,r)Γ(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 Γ(n,3,r)Γ(n,3,r) for p≄5p≄5 , thus extending earlier results for p=2,3p=2,3

    Symmetric-key Corruption Detection : When XOR-MACs Meet Combinatorial Group Testing

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    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

    Birth, growth and computation of pi to ten trillion digits

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    History on the biological nitrogen fixation research in graminaceous plants: special emphasis on the Brazilian experience

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    A note on the generalized Hamming weights of Reed–Muller codes

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    On Repeated-Single-Root Constacyclic Codes

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