Q1(b) · Error control

A transmission error changes one or more bits. A single-bit error flips one bit. A burst error affects a span from the first corrupted bit to the last; the bits in between need not all change. Detection identifies an invalid received pattern; correction identifies the intended pattern within the code’s guaranteed capability.

Block coding

1. Divide data into fixed k-bit datawords. Map each one to an n-bit codeword, with n greater than k.
2. The n−k redundant bits create separation between valid codewords. There are 2^k valid codewords, a subset of the 2ⁿ possible n-bit patterns, for an injective binary block code.
3. A receiver can reject patterns that are not valid codewords. Nearest-codeword decoding can correct errors only if the received pattern has an unambiguous nearest intended codeword under the error bound.
4. Error-control protocols may combine detection with retransmission (ARQ), or use sufficient coding redundancy for forward error correction (FEC).

LaTeX: R_c=k/n,\quad r=n-k