اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTutorial on algebraic deletion correction codes
72
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The deletion channel is known to be a notoriously diffcult channel to design error-correction codes for. In spite of this difficulty, there are some beautiful code constructions which give some intuition about the channel and about what good deletion codes look like. In this tutorial we will take a look at some of them. This document is a transcript of my talk at the coding theory reading group on some interesting works on deletion channel. It is not intended to be an exhaustive survey of works on deletion channel, but more as a tutorial to some of the important and cute ideas in this area. For a comprehensive survey, we refer the reader to the cited sources and surveys. We also provide an implementation of VT codes that correct single insertion/deletion errors for general alphabets at https://github.com/shubhamchandak94/VT_codes/.
قيم البحث
اقرأ أيضاً
Based on the group structure of a unitary Lie algebra, a scheme is provided to systematically and exhaustively generate quantum error correction codes, including the additive and nonadditive codes. The syndromes in the process of error-correction dis
tinguished by different orthogonal vector subspaces, the coset subspaces. Moreover, the generated codes can be classified into four types with respect to the spinors in the unitary Lie algebra and a chosen initial quantum state.
Power decoding is a partial decoding paradigm for arbitrary algebraic geometry codes for decoding beyond half the minimum distance, which usually returns the unique closest codeword, but in rare cases fails to return anything. The original version de
codes roughly up to the Sudan radius, while an improved version decodes up to the Johnson radius, but has so far been described only for Reed--Solomon and one-point Hermitian codes. In this paper we show how the improved version can be applied to any algebraic geometry code.
We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous
paper due to Munuera, Tenorio and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.
We consider families of codes obtained by lifting a base code $mathcal{C}$ through operations such as $k$-XOR applied to local views of codewords of $mathcal{C}$, according to a suitable $k$-uniform hypergraph. The $k$-XOR operation yields the direct
sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the collection of length $k$ walks on an expander graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings on the above hypergraph families. Using known connections between direct sum and direct product, we also recover the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given word, as finding the optimal solution of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code $mathcal{C}$. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the condition and round approach used in several SOS algorithms can then be used to recover the required list of codewords.
Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, ce
rtain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{Omega(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $epsilon$-MSR codes, which incur a factor of $(1+epsilon)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $epsilon$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/epsilon)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
