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\centerline{Mathematics 392 -- Seminar in Coding Theory}
\centerline{Problem Set 2}
\centerline{\it February 1, 2002}
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\noindent
A linear code $C$ is a vector subspace of $K^n$ for some
$n$ (as in our text, $K$ is the binary field $\{0,1\}$ with addition
and multiplication modulo 2).  One of the first advantages
of linear codes is that they let us bring most of the familiar
tools of linear algebra -- vectors, matrices, linear independence,
bases for subspaces and the notion of dimension -- to bear
to study questions about codes.  We will always use the following
notation for linear codes:
$$\cases{ n & is the block length\cr
          k & is the dimension (information bits per codeword)\cr
          d & is the minimum distance\cr}$$
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\noindent
{\it Problems -- due:  in class Friday, February 8}
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\item{A)} From the text:  2.1.1 (and say how you know in each case)
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\item{B)} From the text:  2.1.4
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\item{C)} From the text: 2.3.17
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\item{D)}  For each of the following sets, 
list the elements  of the linear code $C = \langle S \rangle$,
give the parameters $(n,k,d)$ of $C$,  
give a basis of $C$, and find a basis of the dual linear code $C^\perp$. 
\itemitem{1)}  $S = \{0101,1010,1100,0011,1111\}$
\itemitem{2)}  $S = \{11000,01111,11110,01001\}$
\itemitem{3)}  $S = \{10001101,01001000,11001111,01100101\}$
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\item{E)}  From the text: 2.3.22
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\item{F)}  Prove Theorem 2.3.19 from the text.  (Hint: Follow
what you probably did for problem E:  construct
the basis one vector at a time, and count how many choices you 
have at each step.  Then think about how many of your bases
are equal as sets -- recall the order of the elements of a set does
not matter:  $\{a,b,c\} = \{c,b,a\}$, etc.)
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\item{G)} From the text: 2.4.6
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