# The Normalized Second Moment of the Binary Lattice Determined by a Convolutional Code

Published

Journal Article

We calculate 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 modulo 2 binary lattice determined by C. The key observation is that a convolutional code with 2Vstates gives 2Vapproximations 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. In this paper we shall only give details for the convolutional code [1,1 + D], but the method applies to arbitrary codes. We also define the covering radius of a convolutional code, and calculate this quantity for the code [1,1 + D]. © 1994 IEEE

### Full Text

### Duke Authors

### Cited Authors

- Calderbank, AR; Fishburn, PC

### Published Date

- January 1, 1994

### Published In

### Volume / Issue

- 40 / 1

### Start / End Page

- 166 - 174

### Electronic International Standard Serial Number (EISSN)

- 1557-9654

### International Standard Serial Number (ISSN)

- 0018-9448

### Digital Object Identifier (DOI)

- 10.1109/18.272475

### Citation Source

- Scopus