Question
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
Answer :
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A binary code is linear if it contains the all-zero codeword and is closed under addition modulo 2. For (i), the code is (\mathcal{C}={(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)}). The all-zero vector is present. Now check closure under modulo-2 addition. Adding
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Related Questions
Evaluate , where c is the eight like figure shown in Fig. 1.
Find the minimum distance for each of the codes.
Find the constant c such that can be extended to be analytic at
, when
is fixed.
Find the parity check matrix of the code .
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 parity check matrix of the code . Decode the following vectors
i)
ii)
iii)
iv)
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.