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Gents, I have the 128 codewords of a [14,7,4] binary linear code - which is actually the Plotkin $(a\mid a+b)$ construction of $Ham(3,2)$ and $Sim(3,2)$.

Now, I want to have its generating matrix, but it's somehow unclear to me. Any help would be appreciated.

Thanks.

gbiondo
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    Pray, tell us what are $Ham(3,2)$ and $Sim(3,2)$. I am guessing that the first is a Hamming code and the latter is a simplex code. But exactly which codes are they? I suspect that the former is a $[7,4,3]$-code and the latter is its even weight $[7,3,4]$-subcode. But to answer your question we need to define a bit ordering or, a generator matrix. There is no such thing as THE $[7,4,3]$-code. Its bits can come in many an order. True, there are a few more natural ones. The most natural one depends on whether you are an algebraist or an engineer at heart. – Jyrki Lahtonen Jan 28 '15 at 16:18
  • See the comments under this answer where Dilip Sarwate explains the reason for some of the popular bit orders. – Jyrki Lahtonen Jan 28 '15 at 16:39
  • $Ham(3,2)$ is a binary hamming code based on vectors of $\left(\mathbb{Z_2}\right)^3$ and Sim is the simplex code with similar parameters; your surmise was right.

    Assume that the parity-check matrix for the Ham is the one having as column, in order, the binary representation of numbers from 1 to 7 (i.e. - 1st column is 001, second is 010, third is 011, ... last is 111).

    I would say I am a computer scientist at heart ;) anyways I hope this answers

     – gbiondo Jan 28 '15 at 20:08
  • Hint: begin with something like $\displaystyle \left[\begin{matrix}G_1&G_1\0&G_2\end{matrix}\right]$ and then massage things a little to get a systematic generator matrix if that is what your heart desires. – Dilip Sarwate Jan 31 '15 at 14:43

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