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    Coding Theory : Vectors and Generator Matrices

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    1(i) Explain what is meant by
    (a) a linear code over Fq,
    (b) the weight w(u) of a vector u and the distance d(u, v) between vectors u and v.
    (c) Define the weight tu(C) and the minimal distance d(C) of a code C. Prove that w(C) = d(C) if C is linear.
    (ii) Give the definition of the dual code C-'- of a linear code C. Prove that C-'- is
    a linear code.
    (iii) Given a generator matrix C in standard form for a linear code C, describe a generator matrix H in standard form for the dual code C-'-.
    Let C be a linear code over F and let H be a generator matrix for C-'-. Prove ...
    (v) Let C be a 3-ary code with generator matrix
    ...
    (a) Find a generator matrix for C in standard form.
    (b) Find a parity-check matrix for C in standard form.
    (c) Determine d(C). What is your conclusion about the error-correcting and error-detecting powers of the code?
    ....

    1 (i)Exp1ain what is meant by a linear code over Fq, the weight of a vector and the weight of a linear code.
    (b) Let e F. Prove that
    d() =w(?j).
    (ii) Prove that w(C) = d(C) if C is linear.
    (iii) Let D be the linear 3-ary code generated by the matrix
    102
    011
    (a) List the codewords of D.
    (b) Find d(D).
    (c) Write down a standard array for D. Is it possible to give an example of a received vector with one error being correctly decoded?
    (iv) Let C be the code obtained from the code D of part (iii) by adding an extra parity check: be the linear 3-ary code generated by the matrix
    1 0 2 1
    0111?
    Show that C corrects single errors.
    ...

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    https://brainmass.com/math/matrices/coding-theory-vectors-generator-matrices-112280

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    1 (i)Exp1ain what is meant by a linear code over Fq, the weight of a vector and the weight of a linear code

    A linear code over is a linear subspace of a finite dimensional vector space over .
    The weight of a vector is the number its non-zero entries.
    The weight of a code C is the minimum of w(x-y)=weight(x-y) for x,y distinct elements in C.
    Example: ...

    Solution Summary

    Vectors and generator matrices are investigated. The weight of a vector and the distance is determined.

    $2.49

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