I-色≥orfrndm r of（nn decoding rule).This parameter t is called the error correcting capacity of the code. decoding sphered+ The minimum Hamming distance of a linear block code depends on the choice of parity-check equations and the number of parity bits,k 3.Important codes 1)Algebraic block codes -Hamming codes algebraic struct re,algebraic decoding algorithms available. -Golay (23.12)code:A perfect triple-error-correcting code,widely used and generated by g(x)=1+x2+x+x+x+x0+x" -Reed-Muller codes used for error control in data communications and data storages. 2)Convolutional codes:(2,1,6)code generated by g"=(1101101),g2=000111) This code hasd=10 3)Codes(defined)on graphs: Low-density parity-check codes Turbo codes capacity-approching codes 4.Types of error control schemes -Forward-error-correction (FEC):An error-correction code is used -Automatic-repeat-request(ARQ):An error-detection code is used. If the presence of error is detected in a received word,a retransmission is requested.The request signal is sent to the nitter through a feedback channel.Retransmissior continues until no errors being detected -Hybrid ARQ:A proper combination of FEC and ARQ. 5.Decoding8 min 1 2 d t ⎢ ⎥ − = ⎢ ⎥ ⎣ ⎦ or fewer random errors over a block of n digits (using minimum distance decoding rule). This parameter t is called the error correcting capacity of the code. t t min d decoding sphere min d t ≥ + 2 1 „ The minimum Hamming distance of a linear block code depends on the choice of parity-check equations and the number of parity bits, n k − . 3. Important codes 1) Algebraic block codes - Hamming codes - BCH codes: A large class of powerful multiple random error-correcting codes, rich in algebraic structure, algebraic decoding algorithms available. - Golay (23, 12) code: A perfect triple-error-correcting code, widely used and generated by 2 4 5 6 10 11 gx x x x x x x () 1 =+ + + + + + - Reed-Muller codes - Reed-Solomon codes: nonbinary, correcting symbol errors or burst errors ,most widely used for error control in data communications and data storages. 2) Convolutional codes: (2, 1, 6) code generated by (1) (2) g g = = (1101101), (1001111) This code has free d =10 . 3) Codes (defined) on graphs: Low-density parity-check codes capacity-approching codes Turbo codes ⎫ ⎬ ⎭ 4. Types of error control schemes - Forward-error-correction (FEC): An error-correction code is used. - Automatic-repeat-request (ARQ): An error-detection code is used. If the presence of error is detected in a received word, a retransmission is requested. The request signal is sent to the transmitter through a feedback channel. Retransmission continues until no errors being detected. - Hybrid ARQ: A proper combination of FEC and ARQ. 5. Decoding