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### Optimal locally repairable codes of distance $3$ and $4$ via cyclic codes

Like classical block codes, a locally repairable code also obeys the
Singleton-type bound (we call a locally repairable code {\it optimal} if it
achieves the Singleton-type bound). In the breakthrough work of \cite{TB14},
several classes of optimal locally repairable codes were constructed via
subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in
\cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was
proved through extension of construction in \cite{TB14} that length of $q$-ary
optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly,
\cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairable
codes of small distance and locality with code length achieving roughly $q^2$.
Very recently, it was further shown in \cite{LMX17} that there exist $q$-ary
optimal locally repairable codes with length bigger than $q+1$ and distance
propositional to $n$.
Thus, it becomes an interesting and challenging problem to construct new
families of $q$-ary optimal locally repairable codes of length bigger than
$q+1$.
In this paper, we construct a class of optimal locally repairable codes of
distance $3$ and $4$ with unbounded length (i.e., length of the codes is
independent of the code alphabet size). Our technique is through cyclic codes
with particular generator and parity-check polynomials that are carefully
chosen

### Neutronic Analysis on Potential Accident Tolerant Fuel-Cladding Combination U$_3$Si$_2$-FeCrAl

Neutronic performance is investigated for a potential accident tolerant fuel
(ATF),which consists of U$_3$Si$_2$ fuel and FeCrAl cladding. In comparison
with current UO$_2$-Zr system, FeCrAl has a better oxidation resistance but a
larger thermal neutron absorption cross section. U$_3$Si$_2$ has a higher
thermal conductivity and a higher uranium density, which can compensate the
reactivity suppressed by FeCrAl. Based on neutronic investigations, a possible
U$_3$Si$_2$-FeCrAl fuel-cladding systemis taken into consideration. Fundamental
properties of the suggested fuel-cladding combination are investigated in a
fuel assembly.These properties include moderator and fuel temperature
coefficients, control rods worth, radial power distribution (in a fuel rod),
and different void reactivity coefficients. The present work proves that the
new combination has less reactivity variation during its service lifetime.
Although, compared with the current system, it has a little larger deviation on
power distribution and a little less negative temperature coefficient and void
reactivity coefficient and its control rods worth is less important, variations
of these parameters are less important during the service lifetime of fuel.
Hence, U$_3$Si$_2$-FeCrAl system is a potential ATF candidate from a neutronic
view

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