Information sets in cyclic codes

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I am trying to show that in any cyclic code of dimension k, any set of k consecutive coordinates forms an information set. I think the argument can be made by considering the parity check matrix, but I am not sure. Any thoughts?

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Yes. This is true. A plan:

  • Show that it holds for the $k$ last coordinates (Hint: what happens when you use the generator polynomial and do systematic encoding).
  • Apply cyclicity to show that it holds for any $k$ consecutive coordinates (including the shifts that "wrap around").