1/n Turbo codes from linear system point of view

Abstract

The performance of turbo codes at the error floor region is largely determined by the effective free distance, which corresponds to the minimum Hamming weight among all codewords sequences generated by input sequences of weight two. In this paper, we study turbo codes of dimension one obtained from the concatenation of two equal codes and present an upper bound on the effective free distance of a turbo code with these parameters defined over any finite field. We do that making use of the so-called (A, B, C, D) statespace representations of convolutional codes and restrict to the case where A is invertible. A particular construction, from a linear systems point of view, of a recursive systematic convolutional code of rate 1/n so that the effective free distance of the corresponding turbo code attains this upper bound is also presented.

Minimal State-Space Representation of Convolutional Product Codes

“…The blocks are not encoded independently and previously encoded data (matrices in this case) in the sequence have an effect over the next encoded node. Because of this, convolutional codes have memory and can be viewed as linear systems over a finite field (see, for instance, [5,[9][10][11][12][13][14][15][16][17][18]). A description of convolutional codes can be provided by a time-invariant discrete linear system called discrete-time state-space system in control theory (see [19][20][21]).…”