(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose n[itex]\leq[/itex]2

Let C be the code consisting ofall binary stringsof lengthnin which the sum of the bits is even. Is C and MDS code? is C a cyclic code?

2. Relevant equations

An MDS code is one where the codewords are seperated by a maximum number of bitsd. MDS codes obey this theorem: There are q[itex]^{n-d+1}[/itex]different wordsof length n-(d-1) when the alphabet is of size q. I think this means thatif we delete the first (d-1) letters of every codeword of length n,there will be q[itex]^{n-d+1}[/itex] different words of length n-(d-1).

In our case, q = 2 as we are working with binary strings.

e.g. if n = 2, C = {00,11}

n = 3 ---> C = {000,110,101,011}

n = 4 ---> C = {0000,1100,1001,0011,0110,1010,0101,1111}

etc...

3. The attempt at a solution

From observation, d = 2. How do I find this, not simply as an observation? I've checked that the formula fits for n from [2,5] and it seems to.

The codes I have written out appear to be cyclic. Cyclic codes are those where each element can be formed via cyclic shifting of another element.

any hints on how to solidify my argument?

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# MDS/cyclic code, binary string question

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