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 R ec ta ng ul ar C od es • T rip lic at io n co de s: m 1 m 2 m 3 • • • • m 1m 1m 1 m 2m 2m 2 m 3m 3m 3 • • • •R ep ea te d 3 tim es A t r ec ei vi ng e nd , a m aj or ity v ot e is ta ke n. 2 E rr or d et ec tio n an d co rr ec tio n S lid es b as ed o n un kn ow n ou s co nt rib ut or o n th e w eb … 5 3. 4 H am m in g E rr or -c or re ct in g co de s • F in d th e be st e nc od in g sc he m e fo r si ng le -e rr or co rr ec tio n fo r w hi te n oi se . • S up po se th er e ar e m in de pe nd en t p ar ity c he ck s. → It m ea ns n o su m o f a ny c om bi na tio n of th e ch ec ks is a ny o th er c he ck . • E xa m pl e: c he ck 1 : 1 2 5 7 - -- (1 ) ch ec k 2 : 5 7 8 9 -- - (2 ) ch ec k 3 : 1 2 8 9 -- - (3 ) It is n ot in de pe nd en t, be ca us e (1 )+ (2 )= (3 ) • S o th ird p ar ity c he ck p ro vi de s no n ew in fo rm at io n ov er th at o f t he fi rs t t w o, a nd is si m pl y w as te d ef fo rt . 6 • T he s yn dr om e w hi ch r es ul ts fr om w rit in g a 0 fo r ea ch o f t he m pa rit y ch ec ks th at is c or re ct a nd 1 fo r ea ch fa ilu re c an b e vi ew ed a s an m -b it nu m be r an d ca n re pr es en t a t m os t 2 m th in gs . • F or n bi ts o f t he m es sa ge , 2m ≥ n + 1 It is o pt im al w he n m ee ts th e eq ua lit y co nd iti on . ( H am m in g C od es ) • U si ng S yn dr om e to fi nd o ut th e po si tio n of e rr or s. T he id ea l s itu at io n is to u se th e va lu e of S yn dr om e to p oi nt o ut th e po si tio n of e rr or s. 7 • E xa m pl e: 0 0 0 ⇒ no e rr or 0 0 1 ⇒ er ro r ha pp en ed in th e fir st p os iti on