Which of the following binary codes are linear?
i)
ii)
Justify your answer.
A binary code is linear if it contains the all-zero codeword and is closed under addition modulo 2. For (i), the code is ___ ______ ______ _________ ________ ________ _____ _________ ___ _________ _______ __________.
___ _________ __________ ________ __________ _______ ________.
__________ _____ __________ ________ ______ _______ __________.
__________ _____ __________ ________ ________ ____ ___.
________ ____ __________ ___ ________ ______ ____ _________ ____ _______.
__________ __________ ___ __________ _________ ___ ______ ____ __________ __________ _________.
___ ____ __________ _______ ______ ________ ________ _________ _______ ___ _______ ____.
_______ ______ _____ ___ _________ ________ _______ _______ _________ __________ __________ ______.
_________ _________ __________ ________ _______.
_________ ___ _____ __________ ___ _____ ___ ____.
_____ _____ _____ _____ _____ ______ ___ _________ _________.
____ _____ ______ ________ _________ ___ _____ ___ ______ ______ _________.
__________ __________ _________ _____.
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Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
Let and
be two binary codes with generator matrices
respectively.
i) Find the minimum distance of both the codes.
Table 1: Table for F16.
| 0000 | 0 | 1000 | α³ | 1011 | α⁷ | 1110 | α¹¹ |
|---|---|---|---|---|---|---|---|
| 0001 | 1 | 0011 | α⁴ | 0101 | α⁸ | 1111 | α¹² |
| 0010 | α | 0110 | α⁵ | 1010 | α⁹ | 1101 | α¹³ |
| 0100 | α² | 1100 | α⁶ | 0111 | α¹⁰ | 1001 | α¹⁴ |
Find the image of the circle under the mapping
. What happens when
?
he aim of this exercise is to show that every binary repetition code of odd length is perfect.
i) Find the value of t and d for a binary repetition code of length . \hfill (2)
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetition code of odd length is perfect.
) For the binary, (6,3) linear code C with generator matrix
prepare a standard array for decoding. Use it to decode the vectors (1, 1, 1, 0, 1, 1), and (1, 1, 0, 1, 1, 1). \hfill (7)
Find the constant c such that can be extended to be analytic at
, when
is fixed.