Let 𝔽 q be a finite field of q elements and R k = 𝔽 q + u 𝔽 q + … + u k -1 𝔽 q , where u k = 0, k ≥ 2. This article presents generating set for cyclic codes over R k and then provides some criteria to check the reversibility of these codes. For positive integers n and q such that gcd ( n , q ) ≠ 1, there are three presentations for the cyclic codes of length n over R k . We study all these cases and the case gcd ( n , q ) = 1 to obtain necessary and sufficient conditions for reversibility. Further, a relation between reversible cyclic codes and reversible-complement cyclic codes is established that plays a crucial role in DNA computing. Finally, we provide some examples to support our study.
We consider a setup where Alice is transmitting channel coded messages to Bob and intruder Willie is snooping over this communication. Willie does not know the channel code that Alice is using and wishes to identify it in order to decode the intercepted messages. This problem is termed as blind reconstruction of channel codes and has possible applications in cognitive radios and military. In this work, we consider the situation when Willie knows that Alice's channel code belongs to the family of binary cyclic codes and the underlying communication channel is the binary erasure channel. We present an algorithm to identify the parameters of the unknown channel code corresponding to the given data. The key step in our algorithm consists of distinguishing between the two situations when all of the assumed parameters are correct and when either of the parameter is incorrect. As part of analysis, we present a lower bound on probability of correctly distinguishing between these two situations.
As a non-extensive statistical mechanics application, a possible path to generalised information theory is discussed by introducing a family of non-extensive entropies dependent solely on probability: HD±(P). In this scheme, two regimes of probabilities are possible; while the low-probability region exactly coincides with standard information theory, the high-probability regime offers further optimisation in certain information approaches. In this work, we explore two fundamental processes. Firstly, we propose generalisations to Shannon's coding theorems by modifying the ordinary Kraft inequality. This modification will ensure the codes to be uniquely decipherable in the framework of entropies HD±(P). Secondly, we calculate the channel capacity of a binary symmetric channel (BSC) and a binary erasure channel (BEC). Our results suggest an improvement in data compression and transmission with respect to the standard formulation.
In this article, certain linear codes over a finite field are constructed using multiplicative characters of another finite field. We discuss certain sufficient conditions under which these are either LCD codes or linear codes with one-dimensional hull. General Gauss sums are used heavily to achieve these results. Each theorem proved in this paper, is supported by an example with the help of Magma software. Finally, we show that if the base field is of characteristic 2, these codes are isodual.
This paper presents the construction method of composite parity-check (CPC) matrices and a new burst error correcting method for (n, k) Reed-Solomon (RS) codes based on CPC matrices. The proposed method is capable of finding burst error locations by only locations of symbol 0s of syndromes computed by CPC matrices and evaluating the corresponding error magnitudes using CPC matrices. The simulation results show that the proposed method can correct burst errors for the codes with the rate less than about 0.75 with lower complexity and correct burst errors of length up to n - k - 1 and burst erasures of length up to n - k.
