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2. Linear Block Codes Almost all block codes used today belong to a subset called linear block codes. The exclusive OR of two valid codewords creates another valid codeword. Topics discussed in this section: Minimum Distance for Linear Block Codes Simple Parity-Check Code Hamming Code

3. Minimum Hamming Distance The number of 1s in the nonzero valid codeword with the smallest number of 1s. Simple Parity-Check Code A single-bit error-detecting code in which n = k + 1 with d min = 2. The extra bit is selected to make the number of 1s in the codeword even.

4. Table: Simple parity-check code C(5, 4)

5. Figure: Encoder and decoder for simple parity-check code Parity check guaranteed to detect one single error. Find odd number of errors.

6. Figure: Two-dimensional parity-check code

8. Hamming Code Designed with d min = 3. Relationship between m and n in these codes is n = 2 m − 1 and k = n-m. If m=3, then n=7 and k=4, thus C(7,4) with d min = 3. Table: Hamming code C(7,4)

9. Figure: The structure of the encoder and decoder for a Hamming code

10. r 0 = a 2 + a 1 + a 0 modulo-2r 0 = a 2 + a 1 + a 0 modulo-2 r 1 = a 3 + a 2 + a 1 modulo-2r 1 = a 3 + a 2 + a 1 modulo-2 r 2 = a 1 + a 0 + a 3 modulo-2r 2 = a 1 + a 0 + a 3 modulo-2 s 0 = b 2 + b 1 + b 0 + q 0 modulo-2s 0 = b 2 + b 1 + b 0 + q 0 modulo-2 s 1 = b 3 + b 2 + b 1 + q 1 modulo-2s 1 = b 3 + b 2 + b 1 + q 1 modulo-2 s 2 = b 1 + b 0 + b 3 + q 2 modulo-2s 2 = b 1 + b 0 + b 3 + q 2 modulo-2 Generator creates three parity checks are: Checker creates 3-bit syndrome :

11. Table: Logical Decision by Decoder Let us trace the path of three datawords from the sender to the destination: 1. The dataword 0100 becomes the codeword 0100011. 2. The dataword 0111 becomes the codeword 0111001. 3. The dataword 1101 becomes the codeword 1101000.

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