A Tight Lower Bound on the Sub-Packetization Level of Optimal-Access MSR and MDS Codes

The first focus of the present paper, is on lower bounds on the sub-packetization level $\alpha$ of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on $\alpha$ which is shown to be tight for the case $d=(n-1)$ by comparing with recent code constructions in the literature. We also extend our results to an $[n,k]$ MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level $\alpha$ of an MDS code that can carry out repair of any node in a subset of $w$ nodes, $1 \leq w \leq (n-1)$ where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on $\alpha$ for the case of $d=(n-1)$. This bound holds for any $w (\leq n-1)$ and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case $d<(n-1)$. We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of $Q$ nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level $\alpha$. It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code.Comment: Revised for ISIT 2018 submissio

Radiative stability of neutrino-mass textures

Neutrino-mass textures proposed at high-scales are known to be unstable against radiative corrections especially for nearly degenerate eigen values. Within the renormalization group constraints we find a mechanism in a class of gauge theories which guarantees reproduction of any high-scale texture at low energies with radiative stability. We also show how the mechanism explains solar and atmospheric neutrino anomalies through the bimaximal texture at high scale.Comment: 4 pages REVTEX, 1 Postscript fi

Canonical Constraints on Leptonic Cp Violation using UHCR neutrino fluxes

It is shown that one can in principle constrain the CP-violating parameter delta from measurements of four independant |V_{ij}|^2, or three of them and a ratio, in the leptonic sector. To quantify our approach, using unitarity, we derive simple expressions in terms of four independant |V_{ij}|^2 for cos(delta) and an expression for sin^2(delta) from J^2. Thus, depending on the values of |V_{ij}| and their accuracy, we can set meaningful limits on |delta|. To illustrate numerically, if |V_{u1}|^2 is close to 0.1 with a 10% precision, and if |V_{e3}^2 is larger than 0.005 and for values of |V_{e2}|^2 and |V_{u3}|^2 that stay within +-0.1 of the current experimental data leads to a bound pi/2 < |delta| < pi. Alternatively, a certain combination of parameters with values of |V_{e3}|^2 larger than 0.01 leads to a closed bound of 73 < |delta| < 103. In general, we find that it is better to use |V_{u1}|^2 or |V_{t1}|^2 as the fourth independant |V_{ij}|^2 and that over most of the parameter space, delta is least sensitive to |V_{e3}|^2. With just three independant measurements (solar, atmospheric and reactor) it is impossible to set limits on the CP phase. In this respect, we study the use of ultra high energy cosmic (UHCR) neutrino fluxes as the additional fourth information. We find that within the SM, neutrino fluxes of all three flavours will be very similar but that pushing current neutrino data to their extreme values still allowed, ratios of cosmic neutrino fluxes can differ by up to 20%; such large discrepancies could imply negligibly small CP-violation. We also study a non radiative neutrino decay model and find that the neutrino fluxes can differ by a factor of up to 3 within this model and that an accuracy of 10% on the neutrino fluxes is sufficient to set interestin limits on delta.Comment: 8 pages, 2 figures, 5 table