We will use Sugiyama’s algorithm to decode GRS codes. This algorithm uses a modification of the extended Euclidean algorithm to solve the key equation for the ELP and EEP.
Let’s practice using Sugiyama’s algorithm for GRS decoding. For the following problems, we will work with the \([8,4,5]_{13}\) GRS code over \(\F_{13}\) with code locators \(\balpha=(1,4,3,12,9,10,5,8)\) and column multipliers \(\bv=(12,11,2,1,2,11,11,2)\text{.}\) This code has canonical parity check matrix given by
Suppose the vector \(\by=(0,0,0,0,0,0,3,5) \) is received. Step through Sugiyama’s algorithm to find an error vector \(\be\) and decoded codeword \(\bc\text{.}\) You should be able to determine these answers without doing any work at this point in the semester - use this example to solidify your understanding of the process.
Now suppose the vector \(\by=(3,6,0,4,0,5,0,12) \) is received. Step through Sugiyama’s algorithm to find an error vector \(\be\) and decoded codeword \(\bc\text{.}\)
