Properties of Syndrome Distribution for Blind Reconstruction of Cyclic Codes

TL;DR

This paper analyzes the distribution of syndromes in blind reconstruction of binary cyclic codes, revealing conditions for uniformity and intractability issues under noise, thus providing theoretical insights into existing algorithms.

Contribution

It offers a theoretical analysis of syndrome distributions in blind cyclic code reconstruction, including conditions for uniformity and complexity under noise.

Findings

Syndrome distribution can be uniform or restricted uniform in noise-free cases.

Determines necessary and sufficient conditions for syndrome distribution types.

Finding syndrome distribution in noisy cases is generally computationally intractable.

Abstract

In the problem of blind reconstruction of channel codes, the receiver does not have the knowledge of the channel code used at the transmitter and the aim is to identify this unknown channel code corresponding to the given received sequence. In this paper, we study this blind reconstruction problem for binary cyclic codes. In the literature, several researchers have proposed blind reconstruction algorithms that make use of the distribution of the syndromes (remainders) of the received polynomials with respect to a candidate polynomial for the generator polynomial of the cyclic code. However, very limited analysis is available for the syndrome distribution and its properties. In this paper, we study the syndrome structure of the received polynomials. Specifically, we prove that the syndrome distribution of the noise-free sequence can either be uniform or restricted uniform. We also provide the necessary and sufficient conditions for it to be of the either type. For the noise-affected received sequence we prove that, finding the syndrome distribution is in general computationally intractable. We also apply these results to analyze the performance of the existing methods and verify some of the assumptions made in the literature for blind reconstruction.