Forouzan Error Detection and correction Example 10.16 Find the status of the following generators related to two isolated, single-bit errors. a. x+1 b. x4+1 c. x²+x+1 d. x14+x15+1

Status of the Polynomial Generators related to Two Isolated and Single Bit Errors

Before finding the status, important points to remember about polynomial generators are as follows:

• If the generator has more than one term and the coefficient of $ x^0 $ is 1, then all single-bit errors can find out.

• If the generator cannot divide $ x^t +1 $ (t is the value between 0 and n -1), then all isolated double errors can find out.

• A good polynomial generator holds the following characteristics:

  • It should contain at least two terms.

  • It should not divide $ x^t +1 $, for t between 2 and n - 1.

  • It should have the factor of $ x + 1 $.

Status of Polynomial Generators

$ a] x +1 $

This is not a good choice as a generator. Because it cannot detect two errors that are next to each other.

$ b] x^4 + 1 $

This is also not a good choice as a generator. Because it cannot detect two errors that are four positions away.

$ c] x^7 + x^6 + 1 $

This is a perfect choice as a generator. It can find out two isolated and single-bit errors.

$ d] x^{15} + x^{14} + 1 $

This polynomial generator cannot divide $ x^t + 1 $ if the value of t is less than 32,768. A codeword with two isolated errors up to 32,768 bits apart can be found out by using this polynomial generator.