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[2:00pm] S. Venkitesh (IITB)
 Description:
 Combinatorics Seminar
Title: Lift of ReedSolomon code with an application to Nikodym sets
Speaker: S. Venkitesh (IITB)
Date and Time: Feb 14, 2018, 2PM
Venue: Ramanujan Hall, Dept. of Mathematics
Abstract:
We will work over the finite field F_q, q = p^k. The ReedSolomon code
with parameters (q,d), denoted as RS(q,d), is the linear space of all
polynomial functions from F_q to F_q with degree atmost d. The
ReedMuller code with parameters (q,m,d), denoted as RM(q,m,d), is the
mvariable analog of RS(q,d), defined to be the linear space of all
polynomial functions from F_q^m to F_q with total degree atmost d.
A nonempty set N in F_q^m is called a Nikodym set if for every point p
in F_q^m, there is a line L passing through p such that all points on
L, except possibly p, are contained in N. Using the polynomial method
and the code RM(q,m,q2), we can prove the lower bound N >= q^m /
m!. We will outline this proof.
We will then define a new linear code called the mlift of RS(q,d),
denoted as L_m(RS(q,d)), and show that RM(q,m,d) is a proper subspace
of L_m(RS(q,d)). We will use this fact crucially, in a proof very
similar to the earlier one, to obtain the improved lower bound N >=
(1  o(1)) * q^m, when we fix p and allow q to tend to infinity. This
result is due to Guo, Kopparty and Sudan.


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