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The packing radius of C is the largest value of s such that the set of balls of radius s centered at each codeword of C are mutually disjoint. From the proof of the Hamming bound, it can be seen that for = ⌊ ⌋, we have: s ≤ t and t ≤ r.
Since [7, 4, 3] = [n, k, d] = [2 m − 1, 2 m − 1 − m, 3]. The parity-check matrix H of a Hamming code is constructed by listing all columns of length m that are pair-wise independent. Thus H is a matrix whose left side is all of the nonzero n -tuples where order of the n -tuples in the columns of matrix does not matter.
p 1 covers d 1, d 2, d 4; p 2 covers d 1, d 3, d 4; p 3 covers d 2, d 3, d 4; The remaining rows (3, 5, 6, 7) map the data to their position in encoded form and there is only 1 in that row so it is an identical copy. In fact, these four rows are linearly independent and form the identity matrix (by design, not coincidence).
Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit.
Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code words. (The [n, k, d] notation should not be confused with the (n, M, d) notation used to denote a non-linear code of length n, size M (i.e., having M code words), and minimum Hamming distance d.)
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes, turbo codes, and fountain codes.
Specifically, all lines connecting to a variable node (box with an '=' sign) have the same value, and all values connecting to a factor node (box with a '+' sign) must sum, modulo two, to zero (in other words, they must sum to an even number; or there must be an even number of odd values).