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Formally, the set of codewords of the Reed–Solomon code is defined as follows: Since any two distinct polynomials of degree less than agree in at most points, this means that any two codewords of the Reed–Solomon code disagree in at least positions.
A block code can have one or more encoding functions that map messages to codewords . The Reed–Muller code RM (r, m) has message length and block length . One way to define an encoding for this code is based on the evaluation of multilinear polynomials with m variables and total degree at most r. Every multilinear polynomial over the finite field with two elements can be written as follows ...
A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that is divisible by .
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Locally decodable code. A locally decodable code (LDC) is an error-correcting code that allows a single bit of the original message to be decoded with high probability by only examining (or querying) a small number of bits of a possibly corrupted codeword. [1][2][3] This property could be useful, say, in a context where information is being ...
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