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A simplistic example of ECC is to transmit each data bit 3 times, which is known as a (3,1) repetition code. Through a noisy channel, a receiver might see 8 versions of the output, see table below. Through a noisy channel, a receiver might see 8 versions of the output, see table below.
In a polynomial code over () with code length and generator polynomial () of degree , there will be exactly code words. Indeed, by definition, () is a code word if and only if it is of the form () = (), where () (the quotient) is of degree less than .
Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, Reed–Muller codes and Polar codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes ...
RM(m − 1, m) codes are single parity check codes of length N = 2 m, rate = and minimum distance =. RM(m − 2, m) codes are the family of extended Hamming codes of length N = 2 m with minimum distance =. [7]
There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem , any four points can be partitioned into two subsets with intersecting convex hulls , so it is not possible to separate one of these two subsets from ...
[3] [4] In the case where C is a linear subspace of its Hamming space, it is called a linear code. [3] A typical example of linear code is the Hamming code . Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes ...
In a binary alphabet made of ,, if a (,) repetition code is used, then each input bit is mapped to the code word as a string of -replicated input bits. Generally = +, an odd number. The repetition codes can detect up to [/] transmission errors. Decoding errors occur when more than these transmission errors occur.
Many protocols use an XOR-based longitudinal redundancy check byte (often called block check character or BCC), including the serial line interface protocol (SLIP, not to be confused with the later and well-known Serial Line Internet Protocol), [8] the IEC 62056-21 standard for electrical-meter reading, smart cards as defined in ISO/IEC 7816, and the ACCESS.bus protocol.