Thanks for your demonstration but I think the step where you indicated that C' has dimension k-1 is not really obvious. I am not expert enough but there are a couple of notes where I found that they develop a proof for that.
https://www.mathematik.uni-wuerzburg.de/~mueller/Teach/coding.pdf https://www.math.uci.edu/~nckaplan/teaching_files/kaplancodingnotes.pdf — Preceding unsigned comment added by Jsalinas87 (talk • contribs) 22:59, 5 October 2017 (UTC)