# Decoding cyclic code, assuming we have no errors

Assuming the coded data is errorless and given generator polynomial coefficients. By what algorithm can I decode the data coded by matrix constructed by given generator polynomial?

• Have you checked your course material? Or do you mean there should be a faster algorithm assuming the coded data is errorless? Dec 18, 2018 at 20:20
• main problem is that generator matrix is irreversible(as it is not square) so i am confused how to revert coded word to it's original meaning. "assuming the code is errorless" is by problem declaration, that i am trying to solve Dec 18, 2018 at 21:47

Let us start with generator matrix $$G$$, a matrix of $$k$$ rows and $$n$$ columns where $$n\ge k$$. Given any input row vector $$s$$ of $$k$$ characters, we can generate the codeword for $$s$$, a row vector of $$n$$ entries by $$w=sG$$ If given $$s$$, how can we get back $$w$$? Suppose we can find a matrix $$G'$$ of $$n$$ rows and $$k$$ columns such that $$GG'=I_k$$ where $$I_k$$ is the identity matrix of $$k$$, we will have $$s=sI_k=s(GG')=(sG)G'=wG'$$
So the question becomes how to find such matrix $$G'$$ from $$G$$. You should be able to find in your course material the procedure or technique to accomplish that. Note that $$G'$$ is not a square matrix. In general $$G'$$ is not unique, either.
• Do you mean the main problem is how to find the matrix $G'$? Note that $w$ is obtained by multiplying $w$, the coded word with $G'$. There is no "dividing" here. Dec 19, 2018 at 14:17
• We can find $G'$ by an established method that involves no "dividing" by a matrix. Is dividing by a scalar needed? Yes. (In fact, dividing by a scalar can be done by multiplication by a scalar.) Dec 20, 2018 at 10:33
• You should be able to find in your course material the procedure or technique that finds $G'$. Dec 20, 2018 at 10:37