Explaining The Message

Suppose we want to send the message m = (5, 3, 2, 6), and we wish to do this so that we have the ability to detect and correct 1 error if it were to occur. We can use the QRT and send our message as encoded q-relations as well as additional redundant q-relations. If our message contains 2r q-relations and we send 2(r + s) q-relations, then the QRT can correct up to s errors. Thus, we need to send 2 more pieces of redundant data in order to ensure we can correct s = 1 error. We will do this by calculating the q-relations for our original message using the QRT Weave over F7[x] and then using the transform to include two redundant q-relations. 11 Let R = {(5, x − 2),(3, x − 1),(2, x − 5),(6, x − 3)}. Using the QRT Weave provided in Section …show more content…

3. If we find that the q-relations in R0 (α) do not match those in c 0 for only one ˆα ∈ Qˆ then the four q-relations we selected to calculate ˙ hR0i are correct and R0 (Q) is our original message. For example, if we select R0 = {(5, x−2),(3, x−1),(2, x−5),(5, x−3)} then we obtain ˙ hR0i = x 3 + 2x. Performing the QRT on this value, we get R 0 (Q) = {(5, x − 2),(3, x − 1),(2, x − 5),(5, x − 3)(2, x − 4),(4, x − 6)}. Notice that the q-relations in R0 (Q) do not match those in c 0 for 3 q-relations. Hence, there is an error in one of the q-relations selected in R0 . 12 Now, suppose we select R0 = {(5, x−2),(3, x−1),(2, x−5),(6, x−6)}. We obtain ˙ hR0i = 6x 3 + 2x 2 + 3x + 6 and performing the QRT on this value, we get R 0 (Q) = {(5, x − 2),(3, x − 1),(2, x − 5),(6, x − 3)(0, x − 4),(6, x − 6)}. Since the q-relations in R0 (Q) are not the same as those in c 0 for only one q-relation, we can conclude that there is an error in one of the q-relations not selected to calculate ˙ hR0i. This tells us the ˙ hR0i = ˙ hRi and so the q-relations in R0 (Q) form c. We have recovered the original message. In reality, we do not need to preform the entire QRT Weave for every subset of q-relations. Because of the iterative nature of how R¯ is calculated, there is a method of “swapping” these relations for more efficient computation. This technique is outlined in