We develop upper bounds on code size for an independent and identically distributed deletion and insertion channels for a given code length and target frame error probability. The bounds are obtained as a variation of a general converse bound, which, though available for any channel, is inefficient and not easily computable without a good reference distribution over the output alphabet. We obtain a reference output distribution for a general finite-input finite-output channel and provide a simple formula for the converse bound on the capacity employing this distribution. We then evaluate the bound for the deletion channel with a finite block length, and show that the resulting upper bound on the code size is tighter than that for a binary erasure channel, which is the only alternative converse bound for this finite-length setting. We also provide similar results for the insertion channel. Furthermore, we present a simple algorithm for computing an achievability bound for a general discrete-input discrete-output channel. Although the algorithm has exponential complexity, it is useful for comparison purposes.

Particularly motivated by DNA data storage systems, we consider channels with synchronization errors modeled as insertions and deletions, along with substitutions. We focus on the case where the synchronization error process has memory and investigate the information stability of these channels, hence the existence of their Shannon capacity. We assume that the synchronization errors are governed by a stationary and ergodic finite-state Markov chain and prove that such a channel is information-stable, which implies the existence of a coding scheme that achieves the limit of mutual information. This result implies the existence of the Shannon capacity for a wide range of channels with synchronization errors, with implications for various applications.We also provide specific examples of deletion channels with Markov memory and numerically evaluate their capacity bounds, thereby allowing us to quantify the capacity difference between memoryless deletion channels and those with memory, with the same deletion probability, and to reveal that having memory increases the channel capacity.

We develop the first non-asymptotic upper bound on code size for the deletion channel for a given code length and target error probability. The bound is obtained as a variation of a general converse bound, which, though available for any channel, is inefficient and not easily computable without a good reference distribution over the output alphabet. We obtain a reference output distribution for a general finite-input finite-output channel and provide a simple formula for the converse bound on the capacity employing this distribution. We then evaluate the bound for the binary deletion channel with a finite block length and show that the resulting upper bound on the code side is tighter than that for a binary erasure channel, which is the only alternative converse bound for this finite-length setting.

We consider channels with synchronization errors. A classical result for such channels is their information stability, when the synchronization errors are memoryless. In this paper, we extend this result to the case where the synchronization errors have memory. Specifically, we assume that the synchronization errors are governed by a stationary and ergodic finite state Markov chain, and prove that such channel is information-stable, which implies the existence and achievability of the limit of normalized mutual information. This result applies to a wide range of channels with synchronization errors, with different applications including DNA storage. The developed methodology may also be useful to prove other coding theorems for non-trivial channel sequences.

A family of polarizing kernels is presented together with polynomial-complexity algorithm for computing scaling exponent. The proposed convolutional polar kernels are based on convolutional polar codes, also known as b-MERA codes. For these kernels, a polynomial-complexity algorithm is proposed to find weight spectrum of unrecoverable erasure patterns, needed for computing scaling exponent. As a result, we obtain scaling exponent and polarization rate for convolutional polar kernels of size up to 1024.

A lower bound on minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of generalized cosets of the codes generated by bottom rows of the polarizing matrix. Minimum weight of cosets is also used for construction of convolutional polar subcodes, which provide lower FER under list SC decoding compared to polar subcodes and convolutional polar codes.

A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.

An efficient numerically stable successive cancellation decoding algorithm for convolutional polar codes is proposed. The proposed algorithm makes use of only summation and comparison operations, and admits immediate extension to the case of list SC decoding, which provides near-ML performance.