2,137 research outputs found
HFR Code: A Flexible Replication Scheme for Cloud Storage Systems
Fractional repetition (FR) codes are a family of repair-efficient storage
codes that provide exact and uncoded node repair at the minimum bandwidth
regenerating point. The advantageous repair properties are achieved by a
tailor-made two-layer encoding scheme which concatenates an outer
maximum-distance-separable (MDS) code and an inner repetition code. In this
paper, we generalize the application of FR codes and propose heterogeneous
fractional repetition (HFR) code, which is adaptable to the scenario where the
repetition degrees of coded packets are different. We provide explicit code
constructions by utilizing group divisible designs, which allow the design of
HFR codes over a large range of parameters. The constructed codes achieve the
system storage capacity under random access repair and have multiple repair
alternatives for node failures. Further, we take advantage of the systematic
feature of MDS codes and present a novel design framework of HFR codes, in
which storage nodes can be wisely partitioned into clusters such that data
reconstruction time can be reduced when contacting nodes in the same cluster.Comment: Accepted for publication in IET Communications, Jul. 201
Hadamard matrices modulo p and small modular Hadamard matrices
We use modular symmetric designs to study the existence of Hadamard matrices
modulo certain primes. We solve the -modular and -modular versions of
the Hadamard conjecture for all but a finite number of cases. In doing so, we
state a conjecture for a sufficient condition for the existence of a
-modular Hadamard matrix for all but finitely many cases. When is a
primitive root of a prime , we conditionally solve this conjecture and
therefore the -modular version of the Hadamard conjecture for all but
finitely many cases when , and prove a weaker result for
. Finally, we look at constraints on the existence of
-modular Hadamard matrices when the size of the matrix is small compared to
.Comment: 14 pages; to appear in the Journal of Combinatorial Designs; proofs
of Lemma 4.7 and Theorem 5.2 altered in response to referees' comment
Constructions of biangular tight frames and their relationships with equiangular tight frames
We study several interesting examples of Biangular Tight Frames (BTFs) -
basis-like sets of unit vectors admitting exactly two distinct frame angles
(ie, pairwise absolute inner products) - and examine their relationships with
Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one
frame angle.
We demonstrate a smooth parametrization BTFs, where the corresponding frame
angles transform smoothly with the parameter, which "passes through" an ETF
answers two questions regarding the rigidity of BTFs. We also develop a general
framework of so-called harmonic BTFs and Steiner BTFs - which includes the
equiangular cases, surprisingly, the development of this framework leads to a
connection with the famous open problem(s) regarding the existence of Mersenne
and Fermat primes. Finally, we construct a (chordally) biangular tight set of
subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page
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