Rectangular Codes

1
Rectangular Codes

• Triplication codes

m1 m2 m3
Repeated 3 times
m1m1m1 m2m2m2 m3m3m3
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

m -1
o o o o xo o o o xo
o o o x
o o o
o xx x x x x
o message positionx check position
n -1
sum mod 2
Itd better use even-parity checking to avoid
contradiction
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.
• (k21)x(k11) array codeIf 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.

7

• Example 0 0 0 ? no error 0 0 1
  ? error happened in the first position

8

• Locate errorcheck1 m1 m3 m5 m70check2
  m2 m3 m6 m70check3 m4 m5 m6 m70
• Viewing m1 , m2 , m4 as check bit
• Note that the check positions are equally
  corrected with the message positions. The code is
  uniform in its protection. Once encoded there is
  no different between the message and the check
  digits.

9

• Hamming code
• when m 10, then n 1023original message
  length 1023 10 1013Redundancy