I am self studying coding theory from lecture notes and I am unable to prove this result.
Result- If two vectors x, y of $F_{q}$^n are in same coset of C iff they have same syndrome.
My attempt -> Assuming x-y are in same coset it is easy to see that they have same syndrome . But for converse if I assume $ H x^T = H y^T $ , then I am unable to think how it leads to proving that x and y belong to same coset.
Can somebody please help.
Hint: "Coset" here means "coset of $\ \ker(H)\ $", and two vectors $\ v_1, v_2\ $ of a vector space $\ V\ $ are in the same coset of a subspace $\ U\ $ of $\ V\ $ if and only if $\ v_1-v_2\in U\ $.