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Learn how Hamming (7,4) encodes four bits of data into seven bits by adding three parity bits to detect and correct single-bit errors. See the code generator matrix, the parity-check matrix, and the Venn diagram of the overlapping parity coverage.
Hamming codes are a family of linear block codes that can detect and correct one-bit errors, or detect two-bit errors. They are named after Richard W. Hamming, who invented them in 1950 to improve the reliability of punched card readers.
A generator matrix is a matrix whose rows form a basis for a linear code. Learn the definition, format, standard form, equivalence, and parity check matrix of a generator matrix in coding theory.
So for a rth order RM code, we have to decode iteratively r+1, times before we arrive at the final received code-word. Also, the values of the message bits are calculated through this scheme; finally we can calculate the codeword by multiplying the message word (just decoded) with the generator matrix.
A parity-check matrix is a matrix that describes the linear relations of a linear block code. It can be used to check and decode codewords, and its minimum distance is the minimum number of linearly independent columns.
The linear code is defined by its generator matrix, which we choose to be a random generator matrix; that is, a matrix of elements which are chosen independently and uniformly over the field . Recall that in a linear code , the distance equals the minimum weight of a nonzero codeword.
Given the generator matrix of the original Hamming code, the generator matrix for is constructed by taking any two rows from , and is constructed by the remaining two rows of . The corresponding ( 5 × 7 ) {\displaystyle (5\times 7)} parity-check matrix for each sub-code can be generated according to the generator matrix and used to generate ...
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects in finite projective geometry. Let P G ( N , q ) {\displaystyle PG(N,q)} be the finite projective space of (geometric) dimension N {\displaystyle N} over the finite field F q {\displaystyle \mathbb {F} _{q}} .
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