Coding and Cryptography by T. Korner

By T. Korner

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8. List the codewords of the Hamming (7,4) code and its dual. Write down the weight enumerators and verify that they satisfy the MacWilliams identity. 9. (a) Show that if C is linear then so are its extension C +, truncation C ; and puncturing C 0 provided the symbol chosen to puncture by is 0. (b) Show that extension and truncation do not change the size of a code. Show that it is possible to puncture a code without reducing the information rate. (c) Show that the minimum distance of the parity extension C + is the least even integer n with n d(C ).

Show that the n-fold repetition code is perfect if and only if n is odd. 6. Let C be the code consisting of the word 10111000100 and its cyclic shifts (that is 01011100010, 00101110001 and so on) together with the zero code word. Is C linear? Show that C has minimum distance 5. 7. Write down the weight enumerators of the trivial code, the repetition code and the simple parity code. 8. List the codewords of the Hamming (7,4) code and its dual. Write down the weight enumerators and verify that they satisfy the MacWilliams identity.

The rst requirement of a good signature function is that the space M should be much larger than the space S so that H is a many-to-one function (in fact a great-many-to-one function) so that we can not work back from H (M ) to M . The second requirement is that S should be large so that a forger can not (sensibly) hope to hit on H (M ) by luck. 17 During World War II, British bomber crews used to spend the morning before a night raid testing their equipment, this included the radios. 52 Obviously we should aim at the same kind of security as that o ered by our `level 2' for codes:Prospective opponents should nd it hard to nd H (M ) given M if they are in possession of a plentiful supply of message, signature pairs (Mi H (Mi)).

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