V. Non-Binary Codes: Introduction to Reed Solomon Codes

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V. Non-Binary CodesIntroduction to Reed Solomon
Codes
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Reed Solomon Codes

• Reed Solomon Codes are
• Linear Block Codes
• Cyclic Codes
• Non-Binary Codes (Symbols are made up of m-bit
  sequences)
• Used in channel coding for a wide range of
  applications, including
• Storage devices (tapes, compact disks, DVDs,
  bar-codes)
• Wireless communication (cellular telephones,
  microwave links, satellites, )
• Digital television
• High-speed modems (ADSL, xDSL).

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Reed Solomon Codes

• For any positive integer m3, there exists a
  non-binary Reed Solomon code such that

• Code Length n 2m-1
• No. of information symbols k
  2m-1-2t
• No. of parity check symbols n-k 2t
• Symbol Error correcting capability t

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Generator Polynomial

• Example (7,3) double-symbol correcting
  Reed-Solomon Code over GF(23)

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Encoding in Systematic Form

• Three Steps
• Multiply the non-binary message polynomial u(X)
  by Xn-k
• Dividing Xn-ku(X) by g(X) to obtain the remainder
  b(X)
• Forming the codeword b(X)Xn-ku(X)

Encoding Circuit is a Division Circuit
Gate
g2
gn-k-1
g1
g0
..




bn-k-1
b0
b1
b2
Xn-ku(X)
Codeword
Information Symbols
Parity Check Symbols
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Encoding Circuit Example Implementation
Encoding Circuit of (7,3) RS Cyclic Code with
g(X)a3a1X a0X2 a3X3X4
Gate
a3
a0
a3
a1
x
x
x
x




b3
b1
b2
b0
Xn-ku(X)
Codeword
Information Symbols
Parity Check Symbols
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Bits to Symbols Mapping
Receiver Side
Transmitter Side
Decoded Non-Binary Data
Decoded binary Data
Received Non-Binary Data
Binary Data
Non-Binary Data
Encoded Non-Binary Data
Symbol Mapping
Channel Encoder
Channel Decoder
Bit Mapping
Symbol ai(a) Mapping
0 0 0 0 0
a0 a0 1 0 0
a1 a 1 0 1 0
a2 a2 0 0 1
a3 1a 1 1 0
a4 a a2 0 1 1
a5 1a a2 1 1 1
a6 1 a2 1 0 1

• Information message is usually binary
• A symbol mapping is usually necessary for the
  encoding process
• Remember the polynomial representation of symbols
  in GF(2m)

• The coefficients of the polynomial representation
  may be used for the mapping

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Encoding Example

• Binary Information
• Message

0 1 0 1 1 0 1 1 1
Symbol Mapping
Non-Binary Information Message
a a3X a5X2
a5
a3
a

• Encoding Mathematics
• X4u(X) aX4 a3X5 a5X6.
• Dividing by g(X). The remainder b(X) a0 a2X
  a4X2 a6X3
• v(X)b(X)X4u(X) a0 a2X a4X2 a6X3 aX4 a3X5
  a5X6
• V(a0 a2 a4 a6 a a3 a5)
• In Binary V(1 0 0 0 0 1 0 1 1 1 0 1 0 1 0
  1 1 0 1 1 1)

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Encoding Example

• Binary Information
• Message

0 1 0 1 1 0 1 1 1
Symbol Mapping
Non-Binary Information Message
a a3X a5X2
a5
a3
a
Assume u(a a3 a5)
Input Register Contents
0 0 0 0 (Initial State)
a5 a a6 a5 a (1st Shift)
a3 a3 0 a2 a2 (2nd Shift)
a a0 a2 a4 a6 (3rd Shift)
a a3 a5
V(a0 a2 a4 a6 a a3 a5)
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Syndrome Computation

• Valid Codewords V(X) are divisible by g(X)
• a, a2, a3, , a2t are roots of g(X)

Example in (7,3) RS Code r(X) a0 a2X a4X2
a0X3 a6X4 a3X5 a5X6
?r is not a codeword