Review Error correction and error detection What is

• Review

• Error correction and error detection. • What is an error? Bits missed or altered. • How to find an error? – Make sure that when errors occur, the resulting message is invalid. – E. g: two bits message 00, 01, 10, 11 or 00, 11?

• How to find an error? – Introducing redundancy -- using 2 bits message to send 1 bit information in the previous example. – Message = information bits + redundant bits (checksum). • How to design codes that have error correction/detection capability? • Hamming distance between two code words: the number of different bits between the two code words. – E. g 010101 and 111000? Hamming distance = ? • Hamming distance of a complete code: the minimum Hamming distance any of the two codewords in the code. – E. g 010101, 111000, 000111, 111111 Hamming_distance= ?

– Relation between Hamming distance of a code and the code’s error detection/correction capability • Hamming distance = N • How many bits of errors can be detected? • How many bits of errors can be corrected? Example: 0000, 00001111, 11110000, 1111. Hamming distance = ? How many bits of errors can be detected? How many bits of errors can be corrected? Example 2: Parity code (Even parity code)? Let number of information bits = 2, How to construct the even parity code? Hamming distance = ?

– Error correction code: • How many (r) redundant bits do we need to correct a single error for the m information bits? – – A message contains m+r bits total number of possible codewords: 2^(m+r) total number of valid codewords = 2^(m) To correct single error, each single error must results in a different (invalid) codeword. – Total number of (invalid) codewords for one bit error = (m+r)2^m – total number of valid codewords plus the total number of (invalid) codewords for single bit error must be less than the total number of possible codewords. 2^m+(m+r)2^m <= 2^(r+m) m+r+1 <= 2^r

– Error correction code: • How many (r) redundant bits do we need to correct single error for the m information bits? m+r+1 <= 2^r m = 1, m = 2, m = 3, …… m = 1000, r=2 r=3 r = 10 • This formula gives the lower bound of the redundant bits to correct a single error. Hamming code achieves this lower bound (Chapter 3. 2. 1). – How many redundant bits are needed to detect a single error?

– Error correction code/error detection code: which code is more efficient? • Consider only the single bit error: – Assumption: error rate 10^(-6), one error in 1 million bits – packet size 1000 bits for information, total 1000 packets. – Error correction: 1000* (1000+10) – Error detection: 1000*(1000+1) + (1000+1) • Error correction code is in general NOT efficient!!!

• Error detection code: – parity code: detect single error, not good enough. – To detect N errors, we need to have a code whose hamming distance >=N+1. How to get this kind of code? – The most commonly used error detection code is called polynomial code, or cyclic redundancy code or CRC code. A bit stream is treated as a polynomial with coefficients 0 s and 1 s k bits => k-terms polynomial 11001 ---> x^4+x^3+0*x^2+0*x^1+x^0 The sender and the receiver have a generator polynomial G(X) with degree r Add the checksum bits to make sure that the final message is divisible by G(X).

• How to compute the checksum? • Let r be the degree of G(x). Append r zero to the low -order end of data bits (m bits) so that the frame contains m+r bits corresponding to x^r*M(x) • Using modulo 2 division to divide the bit stream corresponding to x^r*M(x) by the bit stream string corresponding to G(x). • Add the remainder back to the frame x^r*M(x). – Example: Data: 1101011011, Generator: 10011 – What would be the final data frame?

• The power of the CRC code – Depends on the selection of the generator, G(x) – T(x): frame polynomial, – E(x) error polynomial – G(x) generator polynomial – T(x) + E(x) / G(x) == E(x) / G(x) – Single error: E(x) = x^j » G(x) has more than two terms, guarantee to detect single error. – Double error: E(x) = x^j+x^k=x^k(x^(j-k)+1) » x^k+1 (k<32768) cannot be divisiable by x^15+x^14+1. – x+1 cannot divide any polynomial with odd number of terms (detect all odd number of errors). – x^r+…+ can detect all less than r number of burst errors.

• Three standard polynomials: – CRC-12 = x^12+x^11+x^3+x^2+x+1 – CRC-16 = x^16+x^15+x^2+1 – CRC-CCITT = x^16+x^12+x^5+1 • What is the hamming distance for the code generated by CRC-16?