Maximum List Size Research Articles

Fast Syndrome-Based Chase Decoding of Binary BCH Codes Through Wu List Decoding

We present a new fast Chase decoding algorithm for binary BCH codes. The new algorithm reduces the complexity in comparison to a recent fast Chase decoding algorithm for Reed–Solomon (RS) codes by the authors (IEEE Trans. IT, 2022), by requiring only a single Kötter iteration per edge of the decoding tree. In comparison to the fast Chase algorithms presented by Kamiya (IEEE Trans. IT, 2001) and Wu (IEEE Trans. IT, 2012) for binary BCH codes, the polynomials updated throughout the algorithm of the current paper typically have a much lower degree. To achieve the complexity reduction, we build on a new isomorphism between two solution modules in the binary case, and on a degenerate case of the soft-decision (SD) version of the Wu list decoding algorithm. Roughly speaking, we prove that when the maximum list size is 1 in Wu list decoding of binary BCH codes, assigning a multiplicity of 1 to a coordinate has the same effect as flipping this coordinate in a Chase-decoding trial. The solution-module isomorphism also provides a systematic way to benefit from the binary alphabet for reducing the complexity in bounded-distance hard-decision (HD) decoding. Along the way, we briefly develop the Gröbner-bases formulation of the Wu list decoding algorithm for binary BCH codes, which is missing in the literature.

On the List Decodability of Insertions and Deletions

In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $\Omega(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $\Omega(1)$ that are list-decodable from a $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions, where $q$ depends only on $\tau_\mathsf{I}$ and $\tau_\mathsf{D}$. We also provide efficient encoding and decoding algorithms for list-decoding from $\tau_\mathsf{I}$-fraction of insertions and $\tau_\mathsf{D}$-fraction of deletions for any $\tau_\mathsf{I} \geq 0$ and $\tau_\mathsf{D} \in [0,1)$. Based on the Johnson-type bound, we derive a Plotkin-type upper bound on the code size in the Levenshtein metric.

Algebraic List-Decoding in Projective Space: Decoding With Multiplicities and Rank-Metric Codes

The problem of list decoding algebraic subspace codes and rank-metric codes is considered. We develop two separate methods, via two different approaches, for list decoding subspace codes and rank-metric codes. These methods provide, for certain code parameters, improved tradeoffs between rate and error-correction capability than that of the Koetter-Kschischang codes, in the domain of subspace codes, and than that of the Gabidulin codes, in the domain of rank-metric codes, and several other extensions thereof. In the first approach, we introduce the notion of root multiplicities for a certain sub-ring of the ring of linearized polynomials. In the list-decoding algorithm, multiple roots are enforced for the interpolation polynomial in order to achieve an improved error-correction radius for an extended family of Koetter-Kschischang subspace codes. The normalized error-correction radius for this approach is τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> =2(L+1)/(r+1)-1-L(L+1)(L+n)/r(r+1)R, where L is the maximum list size, n is the subspace code dimension, R is the rate of the code, and r is the multiplicity parameter. In the second approach, we construct a folded version of Koetter-Kschischang codes. A linear-algebraic list-decoding algorithm is proposed for these codes that achieves the error-correction radius τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> =s(1-sR), where s is the folding parameter. As opposed to the first approach, the size of output list in the second approach depends on the underlying field size and is at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m(s-1)</sup> , where q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is the size of the field that message symbols are chosen from. It is also shown that the output list size is 1, with high probability, in a probabilistic setting. We utilize the techniques of the second approach in the domain of rank-metric codes to construct folded Gabidulin codes to enable a linear-algebraic list-decoding algorithm for such codes. Our algorithm makes it possible to recover.

List Decoding of Insertions and Deletions

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions.

List Decoding of Crisscross Errors

In this paper, list decoding of crisscross errors in arrays over finite fields is considered. For this purpose, the so-called cover metric is used, where the cover of a matrix is a set of rows and columns which contains all non-zero elements of the matrix. A Johnson-like upper bound on the maximum list size in the cover metric is derived, showing that the list of codewords has polynomial size up to a certain radius. Furthermore, a simple list decoding algorithm for a known optimal code construction is presented, which decodes errors in the cover metric up to our upper bound. These results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors.

An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check

In this letter, we propose an adaptive SC (Successive Cancellation)-List decoder for polar codes with CRC. This adaptive SC-List decoder iteratively increases the list size until at least one survival path can pass CRC. Simulation shows that the adaptive SC-List decoder provides significant complexity reduction. We also demonstrate that polar code (2048, 1024) with 24-bit CRC decoded by our proposed adaptive SC-List decoder with very large maximum list size can achieve a frame error rate FER ≤ 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> {-3} at E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> = 1.1dB, which is about 0.25dB from the information theoretic limit at this block length.

Analysis of Verification-Based Decoding on the $q$-ary Symmetric Channel for Large $q$

A new verification-based message-passing decoder for low-density parity-check (LDPC) codes is introduced and analyzed for the q-ary symmetric channel (q-SC). Rather than passing messages consisting of symbol probabilities, this decoder passes lists of possible symbols and marks some lists as verified. The density evolution (DE) equations for this decoder are derived and used to compute decoding thresholds. If the maximum list size is unbounded, then one finds that any capacity-achieving LDPC code for the binary erasure channel can be used to achieve capacity on the q -SC for large q. The decoding thresholds are also computed via DE for the case where each list is truncated to satisfy a maximum list-size constraint. Simulation results are also presented to confirm the DE results. During the simulations, we observed differences between two verification-based decoding algorithms, introduced by Luby and Mitzenmacher, that were implicitly assumed to be identical. In this paper, the node-based algorithms are evaluated via analysis and simulation. The probability of false verification (FV) is also considered and techniques are discussed to mitigate the FV. Optimization of the degree distribution is also used to improve the threshold for a fixed maximum list size. Finally, the proposed algorithm is compared with a variety of other algorithms using both density evolution thresholds and simulation results.

How can good general practitioner care be achieved?

It has been shown that to provide a high standard of care general practitioners probably need to book consultations at intervals of at least 10 minutes. In this study the maximum list size for which a general practitioner might be expected to provide a high standard of care was determined from calculations of the time spent consulting, based on various consultation rates and list sizes and assuming that consultations were 10 minutes long. If good quality care is to be provided and is to include the range of services suggested in the government's recent green paper average list sizes should probably be no more than 1750, and lower in areas of high demand and high need. In addition to this, minimum standards could be determined for such measures as facilities available in surgeries, practice records, and accessibility of doctors to ensure that basic services were offered by all general practitioners.

From the GMSC: Committee wants maximum list size of 2500

From the GMSC: Committee wants maximum list size of 2500