\documentclass[11pt]{amsart}
\usepackage{calc,amssymb,amsmath,fullpage}
\usepackage{enumerate}
\RequirePackage[dvipsnames,usenames]{color}
%\input{kmacros3.sty}
\input{xy}
\xyoption{all}


\def\todo#1{\textcolor{Mahogany}%
{\sffamily\footnotesize\newline{\color{Mahogany}\fbox{\parbox{\textwidth-15pt}{#1}}}\newline}}

\renewcommand{\O}{\mathcal O}
\newcommand{\ie}{{\itshape i.e.} }
\newcommand{\roundup}[1]{\lceil #1 \rceil}
\newcommand{\rounddown}[1]{\lfloor #1 \rfloor}
\renewcommand{\to}[1][]{\xrightarrow{\ #1\ }}
\newcommand{\onto}[1][]{\protect{\xrightarrow{\ #1\ }\hspace{-0.8em}\rightarrow}}
\newcommand{\into}[1][]{\lhook \joinrel \xrightarrow{\ #1\ }}
%\newcommand{DBDual}[1]{{\underline{\omega}_{#1}}}
\newcommand{\DBDual}[1]{{{\uuline \omega}{}^{\mydot}_{#1}}}
\newcommand{\Tt}{{\mathfrak{T}}}
%\newcommand{\bn}{{\mathfrak{n}} }
\newcommand{\sol}[1]{ \vskip 6pt{\color{blue} {\bf Solution:} #1} \vskip 6pt}

\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\va}{{\vec{a}}}
\newcommand{\vb}{{\vec{b}}}
\newcommand{\vzero}{{\vec{0}}}
\newcommand{\vi}{{\vec{i}}}
\newcommand{\vj}{{\vec{j}}}
\newcommand{\vk}{{\vec{k}}}
\newcommand{\vh}{{\vec{h}}}
\newcommand{\vr}{{\vec{r}}}
\newcommand{\vu}{{\vec{u}}}
\newcommand{\vv}{{\vec{v}}}
\newcommand{\vw}{{\vec{w}}}
\newcommand{\vx}{{\vec{x}}}
\newcommand{\vy}{{\vec{y}}}
\newcommand{\vz}{{\vec{z}}}
\newcommand{\vT}{{\vec{T}}}
\newcommand{\vN}{{\vec{N}}}
\newcommand{\vF}{{\vec{F}}}
\newcommand{\vG}{{\vec{G}}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bZ}{\mathbb{Z}}


\begin{document}
\title{Homework \#4 -- Math 5405\\Spring 2016}
\author{Due: Tuesday  4/26/2016}
\maketitle
These are mostly taken from Trappe and Washington.
\begin{enumerate}
\item Two codewords were sent using the Hamming [7,4] code and were received as 0100111 and 0101010.  Each one contains at most one error.  Correct the errors.
\item Let $C$ be the binary code $\{ (0,0,1), (1,1,1), (1,0,0), (0,1,0)\}$.  Show that $C$ is not linear and compute $d(C)$.
\item Let $C \subseteq \mathbb{F}^n$ be a linear code and let $u + C$ and $v + C$ be cosets of $C$.  Show that $u + C = v + C$ if and only if $u - v \in C$.  Further show that every element of $\mathbb{F}^n$ is in exactly one coset.  
\item Let $C$ be the cyclic binary code of length 4 with generating polynomial $g(x) = x^2 + 1$.  Which of the following polynomials correspond to elements of $C$?
\[
1+x+x^3, 1+x+x^2+x^3, x^2 + x^3
\]
\end{enumerate}

\end{document}

