# The linearity of the extended code

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is:

If the code $C$ is linear, can we prove that the extended code $C'$ is linear too?

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With respect to the edit, "last digit" implies add it onto the end, "extra digit" does not. Perhaps the OP can clarify? –  user1729 Apr 12 '13 at 10:14
@ötarcan: Why are you changing this question into something completely different? Even more, since this question already exists as math.stackexchange.com/questions/354151. I did a rollback to the original question (which was addressed with the answers). –  azimut Apr 12 '13 at 12:38

If you add an extra $0$ to every code word, we will have a linear code, with the same minimum distance. Just adding any digit won't give a linear code in general. I suppose if you start with a basis for the code you can add an arbitrary digit to each basis element and then extend linearly to the other code words, and get better distance properties, in principle.

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+1 Most typically the extended symbol is the overall checksum. In other words, after the extension the sum of the components of all the codewords equals zero. This process is a special case of the one described here, and thus preserves linearity. The same can be seen from the fact that calculation of that checksum is a linear function. –  Jyrki Lahtonen Apr 12 '13 at 10:43
to specialize the question;the code C has length n,after adding an extra bit we have the extended code C' that satisfies the following condition:the sum of all the squares of n+1 digit is 0.how can we show the linearness of the extended code that we obtain? –  ötarcan Apr 12 '13 at 11:15
i am so sorry jyrki but i could not understand why this process preserves linearity,can you explain the reason? –  ötarcan Apr 12 '13 at 17:53

A typical extension is to add an overall parity check symbol to $C$, meaning that after the extension, the sum of the entries of any codeword in $C'$ equals $0$. This extended code $C'$ is indeed linear (since the extension rule is linear).

If you just add random symbols, in general $C'$ is not linear.

In general, the extended code C′ is linear if and only if the map $C\to\mathbb F_ q$, mapping a codeword to its extension symbol is linear.

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azimut,do you mean C' is linear if q is even? –  ötarcan Apr 12 '13 at 12:51
@ötarcan: It depends on the selection rule of the extension symbol, as I wrote. –  azimut Apr 12 '13 at 13:20
sorry azimut i could not understand the linearity of the extension rule that was given in the question.which conditions satisfy the linearity of the extended code C'? –  ötarcan Apr 12 '13 at 17:58