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Homework Statement
Suppose n[itex]\leq[/itex]2
Let C be the code consisting of all binary strings of length n in which the sum of the bits is even. Is C and MDS code? is C a cyclic code?
Homework Equations
An MDS code is one where the codewords are separated by a maximum number of bits d. MDS codes obey this theorem: There are q[itex]^{n-d+1}[/itex] different words of length n-(d-1) when the alphabet is of size q. I think this means that if 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...
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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