Narrow Results

In this paper, three families of quantum convolutional codes are constructed. The first one and the second one can be regarded as a generalization of Theorems 3, 4, 7 and 8 [J. Chen, J. Li, F. Yang and Y. Huang, Int. J. Theor. Phys., doi:10.1007/s10773-014-2214-6 (2014)], in the sense that we drop the constraint q ≡ 1 (mod 4). Furthermore, the second one and the third one attain the quantum generalized Singleton bound.

In this paper, a measure of sensitivity is defined to evaluate fault tolerance of neural networks and we show that the sensitivity of a link is closely related to the amount of information passed through it. Based on this assumption, we prove that the distribution of output error caused by s-a-0 (stuck at 0) faults in an MLP network has a Gaussian distribution function. UDBP (Uniformly Distributed Back Propagation) algorithm is then introduced to minimize mean and variance of the output error. Then an MLP neural network trained with UDBP, contributes in an Algorithm-Based Fault Tolerant (ABFT) scheme to protect a nonlinear data process block. A systematic real convolution code guarantees that faults representing errors in the processed data will result in notable nonzero values in syndrome sequence. A majority logic decoder can now easily detect and correct single faults by observing the syndrome sequence. Simulation results demonstrating the error detection and correction behavior against random s-a-0 faults are presented too.

We apply (n, k, m) convolutional codes in which elements in generator matrices are the ones over a finite field to Hybrid ARQ combining with retransmission mechanisms, and evaluate the number of transmitted packets and the number of transmissions. We assume star topology and independent model for a transmission model, and compare two retransmission strategies: (1) one which has been applied with Reed-Solomon (RS) code, and (2) one which choose transmitted packets considering the constraint length m. We use computer simulation for (14, 7, m) and (6, 3, m) convolutional codes to observe the effects of four parameters, that is the number of receivers, constraint length m, packet loss probability, and redundancy at initial transmission PF, to the number of transmissions and transmitted packets. Simulation results showed that the number of transmitted packets can be reduced by the retransmission mechanism which consider m. Also, a PF which minimize number of transmitted packets exists for given packet loss probability, number of receivers, and constraint length.

Maximum distance profile (MDP) convolutional codes have the property that their column distances are as large as possible. It has been shown that, transmitting over an erasure channel, these codes have optimal recovery rate for windows of a certain length. Reverse MDP convolutional codes have the additional advantage that they are suitable for forward and backward decoding algorithms. Beyond that the subclass of complete MDP convolutional codes has the ability to reduce the waiting time during decoding. The first main result of this paper is to show the existence and genericity of $(n,k,\delta )$ complete MDP convolutional codes for all code parameters with $(n-k)|\delta$ as well as that complete MDP convolutional codes cannot exist if $(n-k)\nmid \delta$. The second main contribution is the presentation of two concrete construction techniques to obtain complete MDP convolutional codes. These constructions work for all code parameters with $(n-k)|\delta$ but require that the size of the underlying base field is (sufficiently) large.

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring ${\mathbb{Z}}_{p^r}$. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring ${\mathbb{Z}}_{p^r}$. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated ${\mathbb{Z}}_{p^r}$-modules. We introduce the notion of $b$-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on ${\mathbb{Z}}_p$ is.

Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate $1/2$ and degree $5$ over the field ${𝔽}_{11}$.

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ${\mathbb{Z}}_{p^r}$, ${\mathbb{Z}}_{p^r}((d))$. Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over $𝔽((d))$, where $𝔽$ is a field, when dealing with ${\mathbb{Z}}_{p^r}((d))$ there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called $p$-encoder. In this paper we solve this conjecture for a class of $(2,1)$ convolutional codes over ${\mathbb{Z}}_{p^2}$ and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic $p$-encoder.

We revisit the notion of $r$th generalized column weights for the $j$-truncation of a convolutional code. Taking the limit as $j$ tends to infinity, we define $r$th generalized column weights of a convolutional code. We introduce suitable notions of $j$-equivalence and equivalence, with respect to which generalized column weights are code invariants. We establish some properties of these invariants and compare them with other definitions of generalized distance and weight which appear in the literature.