On Syndrome Decoding of Convolutional Code Cosets

Consider the following problem. Given a syndrome $s$ of a terminated convolutional (or linear block) code $c$ of length $n$ over the field $\mathbb{F}$ with the parity check matrix $H$ and a vector $x\in \mathbb{F}^{n}$, find a vector $e\in \mathbb{F}^{n}$ such that i) it has the syndrome $eH^{T}=5$ and ii) minimizes the distortion $d(x, e^{\backslash }$, where $d(x,\ e)$ is a distance function. This problem is an instance of vector quantization and is called also trellis-coded quantization. Solution of this problem has many practical applications including telecommunication, data compression, and steganography. To solve the problem some publications suggest to use the minimal trellis of the coset and the trellis is designed for every given syndrome, which is complicated. In this paper, we propose using only one trellis of the original code $c$ for all syndromes and a representative $e$ of the coset, where $e$ satisfies the condition i) only. We show a simple method to find a representative $e$ of the coset for every given syndrome $s$.