Maximum Codeword Length Research Articles

AN IMPROVED POLARIZATION WEIGHTS CONSTRUCTION FOR POLAR CODES

Polar codes proposed by Arikan based on channel polarization have been proven to achieve Symmetric Binary-Discrete Input Memoryless Channels capacity. 3GPP has also chosen the Polar codes as the error correction code in the fifth mobile communication system (5G New Radio - 5G NR) with a maximum codeword length of 1024 bits. The design of the code based on the reliability of the synthetic channels has been researched with many different methods to satisfy the requirements demanded in the application of 5G NR. The Polar codes design requirements are ease of implementation, low decoding complexity, and high performance in error correction. This paper has proposed a method of designing Polar codes by determining a synthetic channels reliability index sequence based on the Polarization Weights (PW) method called the Improved Polarization Weights (IPW) method. The simulation results show that the error correction performance of the code designed according to IPW is better than the code structure selected for 5G NR and designed according to the original PW method.

Space-efficient Huffman codes revisited

A canonical Huffman code is an optimal prefix-free compression code whose codewords enumerated in the lexicographical order form a list of binary words in non-decreasing lengths. Gagie et al. (2015) gave a representation of this coding capable of encoding and decoding a symbol in constant worst-case time. It uses σlg⁡ℓmax+o(σ)+O(ℓmax2) bits of space, where σ and ℓmax are the alphabet size and maximum codeword length, respectively. We refine their representation to reduce the space complexity to σlg⁡ℓmax(1+o(1)) bits while preserving the constant encode and decode times. Our algorithmic idea can be applied to any canonical code.

Efficient and compact representations of some non-canonical prefix-free codes

For many kinds of prefix-free codes there are efficient and compact alternatives to the traditional tree-based representation. Since these put the codes into canonical form, however, they can only be used when we can choose the order in which codewords are assigned to symbols. In this paper we first show how, given a probability distribution over an alphabet of σ symbols, we can store an optimal alphabetic prefix-free code in O(σlg⁡L) bits such that we can encode and decode any codeword of length ℓ in O(min⁡(ℓ,lg⁡L)) time, where L is the maximum codeword length. With O(2Lϵ) further bits, for any constant ϵ>0, we can encode and decode O(lg⁡ℓ) time. We then show how to store a nearly optimal alphabetic prefix-free code in o(σ) bits such that we can encode and decode in constant time. We also consider a kind of optimal prefix-free code introduced recently where the codewords' lengths are non-decreasing if arranged in lexicographic order of their reverses. We reduce their storage space to O(σlg⁡L) while maintaining encoding and decoding times in O(ℓ). We also show how, with O(2ϵL) further bits, we can encode and decode in constant time. All of our results hold in the word-RAM model.

Lossless Compression of Intracortical Extracellular Neural Recordings using Non-Adaptive Huffman Encoding.

This paper investigates the effectiveness of four Huffman-based compression schemes for different intracortical neural signals and sample resolutions. The motivation is to find effective lossless, low-complexity data compression schemes for Wireless Intracortical Brain-Machine Interfaces (WI-BMI). The considered schemes include pre-trained Lone 1st and 2nd order encoding [1], pre-trained Delta encoding, and pre-trained Linear Neural Network Time (LNNT) encoding [2]. Maximum codeword-length limited versions are also considered to protect against overfit to training data. The considered signals are the Extracellular Action Potential signal, the Entire Spiking Activity signal, and the Local Field Potential signal. Sample resolutions of 5 to 13 bits are considered. The result show that overfit-protection dramatically improves compression, especially at higher sample resolutions. Across signals, 2nd order encoding generally performed best at lower sample resolutions, and 1st order, Delta and LNNT encoding performed best at higher sample resolutions. The proposed methods should generalise to other remote sensing applications where the distribution of the sensed data can be estimated a priori.

Optimal Merging Algorithms for Lossless Codes With Generalized Criteria

This paper presents lossless prefix codes optimized with respect to a payoff criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new coding algorithm, which transforms the initial source probability vector into a new probability vector according to a merging rule. The coding algorithm is equivalent to a partition of the source alphabet into disjoint sets on which a new transformed probability vector is defined as a function of the initial source probability vector and scalar parameter. The payoff criterion considered encompasses a tradeoff between maximum and average codeword length; it is related to a payoff criterion consisting of a convex combination of average codeword length and average of an exponential function of the codeword length, and to an average codeword length payoff criterion subject to a limited length constraint. A special case of the first related payoff is connected to coding problems involving source probability uncertainty and codeword overflow probability, whereas the second related payoff compliments limited length Huffman coding algorithms.

Incremental Calculation of Minimum-Redundancy Length-Restricted Codes

The length-restricted code-construction problem arises when using prefix codes for large messages and also for search-tree depth minimization. A problem instance comprises an ordered set of input probabilities which in this paper are assumed, without loss of generality, to be a set of unnormalized integer frequencies {f1 , f2,..., fn}, and a maximum codeword length L bits. The package-merge algorithm of Larmore and Hirschberg constructs a minimum-redundancy length-restricted code in O(nL) time. Here we present an algorithm which computes a minimum-redundancy length-restricted code in O((H L + 1)n) time, by starting with a minimum-redundancy (Huffman) code with a maximum codeword length of H, and then refining it to meet the length limit L. The new algorithm is suited to problems where H - L is small; and when H - L > L - [log n], the new algorithm outperforms all previous methods. Experimental results confirm the behavior of the new algorithm

Approximating Huffman codes in parallel

In this paper we present new results on the approximate parallel construction of Huffman codes. Our algorithm achieves linear work and logarithmic time, provided that the initial set of elements is sorted. This is the first parallel algorithm for that problem with the optimal time and work. Combining our approach with the best known parallel sorting algorithms we can construct an almost optimal Huffman tree with optimal time and work. This also leads to the first parallel algorithm that constructs exact Huffman codes with maximum codeword length H in time O ( H ) with n / log n processors, if the elements are sorted.

ALGORITHMS FOR CONSTRUCTION OF OPTIMAL AND ALMOST-OPTIMAL LENGTH-RESTRICTED CODES

In this paper we present new results on sequential and parallel construction of optimal and almost-optimal length-restricted prefix-free codes. We show that length-restricted prefix-free codes with error 1/nk for any k > 0 can be constructed in O(n log n) time, or in O( log n) time with n CREW processors. A length-restricted code with error 1/nk for any k ≤ L/ log Φ n, where [Formula: see text], can be constructed in O( log n) time with n/ log n CREW processors. We also describe an algorithm for the construction of optimal length-restricted codes with maximum codeword length L that works in O(L) time with n CREW processors.

Modified symmetrical reversible variable-length code and its theoretical bounds

Reversible variable length codes (RVLCs) have been adopted in emerging video coding standards-H.263+ and MPEG-4-to enhance their error-resilience capabilities (which are important and essential) in error-prone environments. This study proposes an efficient algorithm to construct a symmetrical RVLC from a given Huffman code. In addition, theoretical bounds on the maximum codeword length for fixed-length Huffman codes, and on the optimal average codeword lengths for sources with exponential distribution are provided.