IV054 Coding, Cryptography and Cryptographic Protocols
2008 ­ Exercises II.
1. How many different binary linear [6, 3] codes C fulfill the condition C = C
?
2. Determine the number of k-dimensional subspaces of the vector space V (n, q).
Explain your reasoning.
3. Let C be a binary code of length n. Consider a binary code C of length n + 1
such that x1    xnxn+1  C if x1 . . . xn  C and xn+1 = n
i=1 xi. Show that if
C is a linear code, then C is also a linear code.
4. Consider the binary linear code C spanned by the codewords 01111, 10111,
11011 and 01100.
(a) Find a generator matrix of C.
(b) Determine how many cosets C has and how many words each of them
contains.
(c) Construct a Slepian array for C and use it to decode 11111 and 00011.
5. Show that the set En of all vectors from Zn
2 which have even weight is a binary
linear code. Find a generator matrix in a standard form for this code.
6. Let C be a binary linear [n, k] code with a parity check matrix H. Consider a
binary code C of length n + 1 such that x1    xnxn+1  C if x1    xn  C and
xn+1 = n
i=1 xi. Show that the matrix
G =
H rT
s 1
,
where r = (00    0) of length n - k and s = (11    1) of length n, is a parity
check matrix of the code C .
7. Show that weight of any non-zero word of code C
, where C = Ham(r, 2), is
2r-1
.
8. Suppose that the matrix
H =


1 1 0 1 0 0
1 0 1 0 1 0
1 1 0 0 0 1


is a parity check matrix of a linear code C. Find the distance of C and the
non-zero word of minimum weight in C.