Description
Let Cin be a code, that is, a block code of length n, dimension k, minimum Hamming distance d, and rate r = k/n, over an alphabet A:
Let Cout be a code over an alphabet B with |B| = |A|k symbols:
The inner code Cin takes one of |A|k = |B| possible inputs, encodes into an n-tuple over A, transmits, and decodes into one of |B| possible outputs. We regard this as a (super) channel which can transmit one symbol from the alphabet B. We use this channel N times to transmit each of the N symbols in a codeword of Cout. The concatenation of Cout (as outer code) with Cin (as inner code), denoted Cout∘Cin, is thus a code of length Nn over the alphabet A:
It maps each input message m = (m1, m2, ..., mK) to a codeword (Cin(m'1), Cin(m'2), ..., Cin(m'N)), where (m'1, m'2, ..., m'N) = Cout(m1, m2, ..., mK).
The key insight in this approach is that if Cin is decoded using a maximum-likelihood approach (thus showing an exponentially decreasing error probability with increasing length), and Cout is a code with length N = 2nr that can be decoded in polynomial time of N, then the concatenated code can be decoded in polynomial time of its combined length n2nr = O(N⋅log(N)) and shows an exponentially decreasing error probability, even if Cin has exponential decoding complexity. This is discussed in more detail in section Decoding concatenated codes.
In a generalization of above concatenation, there are N possible inner codes Cin,i and the i-th symbol in a codeword of Cout is transmitted across the inner channel using the i-th inner code. The Justesen codes are examples of generalized concatenated codes, where the outer code is a Reed–Solomon code.
Read more about this topic: Concatenated Error Correction Code
Famous quotes containing the word description:
“As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
—Henry David Thoreau (1817–1862)
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)
“Once a child has demonstrated his capacity for independent functioning in any area, his lapses into dependent behavior, even though temporary, make the mother feel that she is being taken advantage of....What only yesterday was a description of the child’s stage in life has become an indictment, a judgment.”
—Elaine Heffner (20th century)