Rectangular Codes and Hamming Error-correcting Codes, Study notes of Computer Architecture and Organization

Download Rectangular Codes and Hamming Error-correcting Codes and more Study notes Computer Architecture and Organization in PDF only on Docsity! 1 1 Rectangular Codes • Triplication codes: m1 m2 m3 • • • • m1m1m1 m2m2m2 m3m3m3 • • • • Repeated 3 times At receiving end, a majority vote is taken. 2 Error detection and correction Slides based on unknown ous contributor on the web… 3 •Rectangular codes: Redundancy: o o o • • • o x o o o • • • o x o o o • • • o x • • • • • • • • • • • • o o o • • • o x x x x x x m -1 n -1 o = message position x = check position It’d better use even-parity checking to avoid contradiction sum mod 2 )1)(1( 1 )1( 1 )1( 1 1 −− + − + − + nmnm ) )1)(1( 1)1()1()1)(1( )1)(1( ( −− +−+−+−−= −− nm nmnm nm mn∵ 4 • For a given size mn, the redundancy will be smaller the more the rectangle approaches a square. • For square codes of size n ,we have (n -1)2 bits of information. And 2n-1 bits of checking along the sides. • Note that: Rectangular codes also can correct bursty error. • (k2+1)x(k1+1) array code If k2 ≥ 2(k1-1) ⇒ we can correct k1 size of bursty errors 5 3.4 Hamming Error-correcting codes • Find the best encoding scheme for single-error correction for white noise. • Suppose there are m independent parity checks. →It means no sum of any combination of the checks is any other check. • Example: check 1 : 1 2 5 7 --- (1) check 2 : 5 7 8 9 --- (2) check 3 : 1 2 8 9 --- (3) It is not independent, because (1)+(2)=(3) • So third parity check provides no new information over that of the first two, and is simply wasted effort. 6 • The syndrome which results from writing a 0 for each of the m parity checks that is correct and 1 for each failure can be viewed as an m-bit number and can represent at most 2m things. • For n bits of the message, 2m ≥ n + 1 It is optimal when meets the equality condition. ( Hamming Codes ) • Using Syndrome to find out the position of errors. The ideal situation is to use the value of Syndrome to point out the position of errors.