Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes

In this work, we present hardware and software implementations of flexible polar systematic encoders and decoders. The proposed implementations operate on polar codes of any length less than a maximum and of any rate. We describe the low-complexity, highly parallel, and flexible systematic-encoding algorithm that we use and prove its correctness. Our hardware implementation results show that the overhead of adding code rate and length flexibility is little, and the impact on operation latency minor compared to code-specific versions. Finally, the flexible software encoder and decoder implementations are also shown to be able to maintain high throughput and low latency.Comment: Submitted to IEEE Transactions on Communications, 201

An improved algorithm of generating shortening patterns for polar codes

The rate matching in polar codes becomes a solution when non-conventional codewords of length N≠2n are required. Shortening is employed to design arbitrary rate codes from a mother code with a given rate. Based on the conventional shortening scheme, length of constructed polar codes is limited. In this paper, we demonstrate the presence of favorable and unfavorable shortening patterns. The structure of polar codes is leveraged to eliminate unfavorable shortening patterns, thereby reducing the search space. We generate an auxiliary matrix through likelihood and subsequently select the shortening bits from the matrix. Unlike different existing methods that offer only a single shortening pattern, our algorithm generates multiple favorable shortening patterns, encompassing all possible favorable configurations. This algorithm has a reduced complexity and suboptimal performance, effectively identifying shortening patterns and sets of frozen symbols for any polar code. Simulation results underscore that the shortened polar codes exhibit performance closely aligned with the mother codes. Our algorithm addresses this security concern by making it more difficult for an attacker to obtain the information set and frozen symbols of a polar code. This is done by generating multiple shortening patterns for any polar code

Constructions of Rank Modulation Codes

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. For instance, from binary BCH codes we obtain codes correcting $t$ Kendall errors in $n$ memory cells that support the order of $n!/(\log_2n!)^t$ messages, for any constant $t= 1,2,...$ We also construct families of codes that correct a number of errors that grows with $n$ at varying rates, from $\Theta(n)$ to $\Theta(n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the trade-off between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate. Finally, we list a number of possibilities for constructing codes of finite length, and give examples of rank modulation codes with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor