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Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding [article]

Hannes Bartz, Sven Puchinger
2021 arXiv   pre-print
Theory, 2019) showed that lifted linearized Reed-Solomon codes are suitable codes for error control in multishot network coding.  ...  Although our best upper bound on the list size is exponential, we present a heuristic argument and simulation results that indicate that the list size is in fact one for most channel realizations up to  ...  The set of skew polynomials with coefficients in F q m together with ordinary polynomial addition and the multiplication rule xa = σ(a)x, a ∈ F q m (2) forms a non-commutative ring denoted by F q m [x,  ... 
arXiv:2101.05604v4 fatcat:rkjvgtjzibhdzcrrqlsjs2kppe

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric [article]

Hannes Bartz and Thomas Jerkovits and Sven Puchinger and Johan Rosenkilde
2021 arXiv   pre-print
polynomial rings.  ...  problems related to the remainder-evaluation of skew polynomials.  ...  bases over skew polynomial rings.  ... 
arXiv:2005.09916v2 fatcat:sqtkw73frraqhfiol7evekjcwy

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric [article]

Felicitas Hörmann, Hannes Bartz
2022 arXiv   pre-print
The presented decoder can be used for either list or probabilistic unique decoding and requires at most 𝒪(sn^2) operations in 𝔽_q^m, where s ≤ h is an interpolation parameter and n denotes the length  ...  Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography.  ...  The interpolation system (9) can be solved using skew Kötter interpolation from [10] (similar as in [5, Sec. V]) requiring at most O(sn 2 ) operations in F q m . Lemma 2 (Existence).  ... 
arXiv:2109.14943v2 fatcat:dd3yzncugffshmf7oq476peewi

Sub-Quadratic Decoding of Gabidulin Codes [article]

Sven Puchinger, Antonia Wachter-Zeh
2016 arXiv   pre-print
In particular, we present fast algorithms for division, multi-point evaluation and interpolation of linearized polynomials and show how to efficiently compute minimal subspace polynomials.  ...  This paper shows how to decode errors and erasures with Gabidulin codes in sub-quadratic time in the code length, improving previous algorithms which had at least quadratic complexity.  ...  polynomial division to skew polynomial multiplication.  ... 
arXiv:1601.06280v2 fatcat:skvl6rptjzgzdn3z7jk7zrmt4i

Row Reduction Applied to Decoding of Rank Metric and Subspace Codes [article]

Sven Puchinger and Johan Rosenkilde né Nielsen and Wenhui Li and Vladimir Sidorenko
2016 arXiv   pre-print
We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding ℓ-Interleaved Gabidulin codes.  ...  Inspired by row reduction of [x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form.  ...  Related Work In this paper we consider skew polynomial rings over finite fields without derivations [37] (see Section 2.1 for this restricted definition of skew polynomials).  ... 
arXiv:1510.04728v3 fatcat:67v6yro6czegbivydc72xh36f4

Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases

Hannes Bartz, Antonia Wachter-Zeh
2018 Advances in Mathematics of Communications  
An efficient procedure for constructing the required minimal Gröbner basis using the general Kötter interpolation is presented.  ...  The scheme can be used as a (not necessarily polynomial-time) list decoder as well as a polynomial-time probabilistic unique decoder.  ...  for the interpolation module. Modules over polynomial rings are a generalization of the concept of vectors spaces, where the corresponding vector entries and the scalars are from a polynomial ring.  ... 
doi:10.3934/amc.2018046 fatcat:xekbulh5gnam5f3ww5hfuhgmke

Row reduction applied to decoding of rank-metric and subspace codes

Sven Puchinger, Johan Rosenkilde né Nielsen, Wenhui Li, Vladimir Sidorenko
2016 Designs, Codes and Cryptography  
We apply this to solve generalised shift register problems, or Padé approximations, over skew polynomial rings, which occur in error and erasure decoding -Interleaved Gabidulin codes.  ...  Inspired by row reduction of F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into certain normal forms.  ...  which we used in [31] . 2 Skew polynomial rings can be considered having non-zero derivation [48] .  ... 
doi:10.1007/s10623-016-0257-9 fatcat:cjjywbsrz5c7nd457vcccn5wgm

Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

Umberto Martinez-Penas, Frank R. Kschischang
2019 IEEE Transactions on Information Theory  
Multishot network coding is considered in a worst-case adversarial setting in which an omniscient adversary with unbounded computational resources may inject erroneous packets in up to t links, erase up  ...  Combined with the obtained field size, the given decoding complexity is of O(n^' 4ℓ^2 (ℓ)^2) operations in F_2 .  ...  An algorithm with quadratic complexity based on Kötter interpolation was given in [58, Sec. 4] .  ... 
doi:10.1109/tit.2019.2912165 fatcat:qziaieilgvevddundhpyqpg6ty

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants [article]

Hannes Bartz, Sven Puchinger
2022
The results are extended to decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew metric.  ...  and skew metric.  ...  . ≺ w can be computed efficiently using the multivariate Kötter interpolation over skew polynomial rings from [64] , [65] .  ... 
doi:10.48550/arxiv.2201.01339 fatcat:ju5xwlp6cfgbtd4y5qli5gz7eu

Introduction to JINS Special Issue on Human Brain Connectivity in the Modern Era: Relevance to Understanding Health and Disease

Deanna M. Barch, Mieke Verfaellie, Stephen M. Rao
2016 Journal of the International Neuropsychological Society  
After realignment, volumes with movement or artifacts greater than 1.5 mm over three volumes or less were interpolated. Only one subject required interpolation.  ...  These trends are easy to address by simply regressing a linear (or higher) polynomial fit from the timeseries at each voxel. dMRI Diffusion MRI (dMRI) is unique in its ability to map, in vivo, the axonal  ...  Furthermore, the cumulative distribution of the number of reduced connections in TBI over controls skewed toward long-range connections ( Figure 3A) , and notable disruptions occurred in inter-hemispheric  ... 
doi:10.1017/s1355617716000047 fatcat:f2preenihbes5ftkrxbo7tgt64