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The normalized second moment of the binary lattice determined by a convolutional code

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2 Author(s)
Calderbank, A.R. ; AT&T Bell Labs., Murray Hill, NJ, USA ; Fishburn, P.C.

Calculates the per-dimension mean squared error μ(S) of the two-state convolutional code C with generator matrix [1,1+D], for the symmetric binary source S=(0,1), and for the uniform source S={0,1}. When S=(0,1), the quantity μ(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S={0,1}, the quantity μ(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2υ states gives 2υ approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code [1,1+D], but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code [1,1+D]

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Information Theory, IEEE Transactions on  (Volume:40 ,  Issue: 1 )