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\begin{document}
\noindent {\large Essential Coding Theory \hfill
Madhu Sudan\\[-.1in]
CS 229r - Spring 2017 \hfill \mbox{}\\[-.1in]
Due: Wednesday, February 22, 2017\hfill \mbox{}\\[.4in]}
{\LARGE \centering Problem Set 2 \\[.4in] \par}
\section*{Instructions}
\begin{description}
\item[Collaboration:] Collaboration is allowed and encouraged, but you must write everything up by yourselves. You must list all your collaborators.
\item[References:]
Consulting references is OK. However,
in general, try not to run to reference material to answer
questions. Try to think about the problem to see if you can
solve it without consulting {\em any} external sources. If this fails,
you may look up any reference material.
Cite all references (in addition to listing all collaborators).
Explain why you needed to consult any of the references,
if you did consult any.
\item[``Not to be turned in'' Problems]. If a problem is marked ``not to be
turned in'', you don't have to. But you must include a statement saying you know
how to solve the problem. If you are unsure, write up your solution and turn it
in. These problems won't count for your grades, but are good review exercises.
\item[Submission:] Submit your solutions on Canvas. If you do not have a canvas
account, or are not taking this course for credit, but want to submit solutions
to the psets anyway, please contact me (madhu) right away and I'll try to find a
solution for you.
\end{description}
\begin{verbatim}
\end{verbatim}
\section*{Problems}
\begin{enumerate}
\item {\bf (Need not be turned in)} {\sf (Finite field as vector spaces)}
\begin{enumerate}
\item Let $q$ be a prime power and $t$ be a positive integer and let $\F_{q^t}$
be a field of size $q^t$. Show that there is a unique copy of $\F_q$ contained in
$\F_{q^t}$, i.e., $q$ elements that are closed under addition and multiplication
and this form a field. (This allows us to talk about multiplication of an element
of $\F_q$ with an element of $\F_{q^t}$ below.)
\item Show that there is
a bijection $\Phi: \F_{q^t} \to \F_q^t$ that is $\F_q$-linear (i.e., for every
$\alpha, \beta \in \F_q \subseteq \F_{q^t}$ and $\gamma,\delta \in \F_{q^t}$ it
is the case that $\Phi(\alpha \gamma + \beta \delta) = \alpha\Phi(\gamma) +
\beta\Phi(\delta)$).
\item Conclude that if $C \subseteq \F_{q^t}^n$ is a linear code over
$\F_{q^t}$ then $C \circ \Phi \subseteq \F_{q}^{tn}$ is a linear code over
$\F_q$, where $C \circ \Phi = \{ (\Phi(u_1),\ldots,\Phi(u_n)) | (u_1,\ldots,u_n)
\in C\}$.
\end{enumerate}
\item {\sf (GV bound by Partitioning)} Recall that linear codes $C_1,\ldots,C_M
\subseteq \{0,1\}^n$ form a {\em partition} if (i) All codes are of the same size (soe
$|C_i| = |C_j|$ for all $i,j \in [M]$); (ii) They cover $\{0,1\}^n$, i.e.,
$\cup_{i\in [M]} C_i = \{0,1\}^n$; and (iii) They have the minimal possible
intersection i.e., $C_i \cap C_j = \{0^n\}$ for all $i \ne j$.
\begin{enumerate}
\item Prove that if $C_1,\ldots,C_M$ form a partition, and $d$ is such that
$\sum_{i=1}^{d-1} {n \choose i} < M$ then there exists $i \in [M]$ such that
$\Delta(C_i) \geq d$.
\item (The Wozencraft ensemble): Let $n = 2t$, and let $\Phi$ be linear bijection from $\F_{2^t}$ to
$\F_2^t$. For $\alpha \in \F_{2^t}$ let $C_{\alpha} \subseteq \F_{2^t}^2$ be the
code with codewords $\{(\beta,\alpha\cdot \beta) | \beta \in \F_{2^t}\}$
and $\F_\infty = \{(0,\beta) | \beta \in \F_{2^t}\}$.
Further, let $\tilde{C}_\alpha = C_\alpha \circ \Phi$. Show that
$\{\tilde{C}_\alpha\}_{\alpha \in \F_{2^t}\cup \{\infty\}} \subseteq \{0,1\}^n$ form a partition.
Conclude that for infinitely many $n$ there is a code of length $n$ with rate
$1/2$ and relative distance approaching $H^{-1}(1/2)$.
\item Prove that for most $\alpha$, the code $C_{\alpha}$ has relative distance
approaching $H^{-1}(1/2)$.
\item Extend Part (b) to get codes of rate $1/\ell$ and distance
approaching $H^{-1}(1 - 1/\ell)$ for (constant) every positive integer $\ell$.
\item (Extra credit:) Extend the notion of a partition to a notion of `uniform
cover'' so as to build codes of rate $1 - 1/\ell$ and distance $H^{-1}(1/\ell)$.
\end{enumerate}
\item {\sf ($q$-ary Plotkin bound)}. The $q$-ary Plotkin bound says that if
$\{C_i = (n_i,k_i,d_i)_q\}$ is an infinite family of codes with $k_i/n_i \geq R$
and $d_i/n_i \geq \delta$ then $R \leq 1 - (q/(q-1))\delta$. Your goal is to
prove this bound below.
\begin{enumerate}
\item Prove that there exist vectors $\eta_1,\ldots,\eta_q \in \R^{q-1}$ such
that $\langle \eta_i,\eta_j \rangle = 1$ if $i=j$ and $-1/(q-1)$ otherwise.
\item Use the above to show that if $c_1,\ldots,c_K \in [q]^n$ have pairwise
distance at least $(q-1)/q \cdot n$, then $K \leq 2(q-1)n$. (Conclude that if
$\delta \geq (q-1)/q$ then $R = 0$.)
\item Show that if there exists an $(n,k,d)_q$ code then there exists an
$(n-1,k-1,d)_q$ code.
\item Combine the above to infer the Plotkin bound.
\end{enumerate}
\item {(\bf Need not be turned in.)} {\sf (The Johnson bound.)}
\begin{enumerate}
\item Prove that if $w,c_1,\ldots,c_L \in \{0,1\}^n$ are words such that
$\Delta(w,c_i) \leq \tau n$ and $\Delta(c_i,c_j) \geq \delta n$ for every $i \ne
j$ and $\tau = 1/2(1 - \sqrt{1 - 2\delta})$ then $L \leq 2n$. (Hint: You may use
the usual transformation to vectors in $\{-1,+1\}^n$ and convert statements about
Hamming distance into statements about inner products. Suppose the vector $w$ is
transformed to $W \in \{-1,+1\}^n$ this way and
vectors $c_1,\ldots,c_L$ to $C_1,\ldots,C_L$. Show that under the given condition on $\tau$ and $\delta$, you can ``shift'' the
origin to some point $\alpha W$ (for $\alpha\in [0,1]$) such that the from the
new origin end points of the vectors $C_1,\ldots,C_L$ have pairwise
non-positive inner product.)
\item Prove the $q$-ary version of the above.
\item Prove that your bound on the relationship between $\tau$ and $\delta$ is
tight.
\end{enumerate}
\item {\sf (Concatenated codes and Justesen codes).}
\begin{enumerate}
\item Use the Wozencraft ensemble to construct, in time $\poly(n)$, binary
codes of length $n$, rate $1/4$ of relative distance approaching $1/2
(H^{-1}(1/2))$. (Your code should be linear and your algorithm should construct the
entire generator matrix in polynomial time.)
\item Now we will show that concatenating with an ensemble of codes also works almost as
well: Let $C_1,\ldots,C_N$ be $(n,k,d)_2$ codes such that most ($N - o(N)$) codes have
distance $d - o(d)$. Suppose $B$ is in $(N,K,D)_{2^k}$ code. Show that the
concatenation of $B$ with $(C_1,\ldots,C_N)$, which is obtained by encoding a
message of $B$ by the encoder of $B$, and then encoding the $i$th symbol of the
encoding by the encoder for $C_i$, is a code of distance $D\cdot d - o(N \cdot
d)$.
\item Conclude that there is a strongly explicit code, one which has a generator
matrix $G = [G_{ij}]$ with $G_{ij}$ being computable in time $\poly\log(n)$ given
$i$ and $j$, of rate $1/4$ with relative distance approaching $1/2 H^{-1}(1/2)$.
\end{enumerate}
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\end{enumerate}
\end{document}