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# Coding Theory : Linear Codes

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Please see the attached file for the fully formatted problems.
(a) Explain what is meant by
(i) a linear code over Fq
(ii) the weight w(x) of a vector x
(iii) the weight w(C) of a code. Prove that,...

(b) Prove that w(C) = d(C) if C is a linear code.

(c) Define F-linear equivalence of codes. State the three row and two column operations which may be performed on generator matrices to obtain F-linearly equivalent codes, and prove that these operations do preserve F-linear equivalence.
(d) Let C be the linear ]F3-code with generator matrix
....
(1) List the codewords of C.
(ii) Find d(C). How many errors can it detect? How many errors can it correct?
(iii) Prove that C is a perfect code.

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Let C be the linear -code with generator matrix

Prove that C is a perfect code.

The ...

#### Solution Summary

Linear codes are investigated.

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