6 research outputs found
Asymptotically Good Additive Cyclic Codes Exist
Long quasi-cyclic codes of any fixed index have been shown to be
asymptotically good, depending on Artin primitive root conjecture in (A.
Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result
to construct good long additive cyclic codes on any extension of fixed degree
of the base field. Similarly self-dual double circulant codes, and self-dual
four circulant codes, have been shown to be good, also depending on Artin
primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and (
M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent
results, we can show that long cyclic codes are good over \F_q, for many
classes of 's. This is a partial solution to a fifty year old open problem
Functional Kuppinger-Durisi-B\"{o}lcskei Uncertainty Principle
Let be a Banach space. Let and ,
satisfy for all
, for all . If is such that , then we show that \begin{align}\label{FKDB} (1)
\quad\quad\quad\quad \|\theta_fx\|_0\|\theta_gx\|_0\geq
\frac{\bigg[1-(\|\theta_fx\|_0-1)\max\limits_{1\leq j,r \leq n,j\neq
r}|f_j(\tau_r)|\bigg]^+\bigg[1-(\|\theta_g x\|_0-1)\max\limits_{1\leq k,s \leq
m,k\neq s}|g_k(\omega_s)|\bigg]^+}{\left(\displaystyle\max_{1\leq j \leq n,
1\leq k \leq m}|f_j(\omega_k)|\right)\left(\displaystyle\max_{1\leq j \leq n,
1\leq k \leq m}|g_k(\tau_j)|\right)}. \end{align}
We call Inequality (1) as \textbf{Functional Kuppinger-Durisi-B\"{o}lcskei
Uncertainty Principle}. Inequality (1) improves the uncertainty principle
obtained by Kuppinger, Durisi and B\"{o}lcskei \textit{[IEEE Trans. Inform.
Theory (2012)]} (which improved the Donoho-Stark-Elad-Bruckstein uncertainty
principle \textit{[SIAM J. Appl. Math. (1989), IEEE Trans. Inform. Theory
(2002)]}). We also derive functional form of the uncertainity principle
obtained by Studer, Kuppinger, Pope and B\"{o}lcskei \textit{[EEE Trans.
Inform. Theory (2012)]}.Comment: 9 Pages, 0 Figure