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I have been thinking of several properties which could distinguish binary codes from non binary codes. However I'm not sure I'm right, so please tell me your opinion:

  • The $(u|u+v)$ construction of Plotkin (1960): Given two codes $C_1$, $C_2$ with parameters $(n,M_i,d_i)$ we construct a new code $C_3$, containing all words that start with $u \in C_1$ and then comes the word $u+v$ where $v \in C_2$ and addition is done by coordinates (modulus the number $q$ of letters we use). If these are binary codes ($q=2$), we get a code with parameters $(2n,M_1M_2,\min\{2d_1,d_2\})$. Does it hold if $q \neq 2$? I tend to think it doesn't but cannot find a counter example.
  • Given a $q$-ary code $C$, we can list the differences between its different words - listing that there are $n_1$ pairs of words with $d=1$, $n_2$ words that differ on two characters and so on (Hamming distance). Given two codes with the same list, does it imply they are equivalent? It is not very difficult to find a counter example if we use ternary codes, but does it hold for binary codes?
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    Oh, I just confirmed the inequality. I don't have the book (I have _A First Course in Coding Theory_). However looking at their solution I think it can be modified using this notation. I'll have a closer look later; thank you! Do you have any idea regarding the second thought?2012-07-15

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