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On the algebraic representation of selected optimal non-linear binary codes [article]

Marcus Greferath, Jens Zumbrägel
2012 arXiv   pre-print
Revisiting an approach by Conway and Sloane we investigate a collection of optimal non-linear binary codes and represent them as (non-linear) codes over Z4.  ...  The Fourier transform will be used in order to analyze these codes, which leads to a new algebraic representation involving subgroups of the group of units in a certain ring.  ...  ACKNOWLEDGEMENTS The authors would like to thank Michael Kiermaier and Oliver Gnilke for inspiring discussions.  ... 
arXiv:1109.4770v2 fatcat:n6qw2y53brcytfin7usbigymuq

Page 7944 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
The set of encoding rules for this code is a representation system for cosets of a certain subgroup in the projective transformation group.  ...  (E-BALE; Palma) Variations on minimal codewords in linear codes. (English summary) Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), 96-105, Lecture Notes in Comput.  ... 

Page 6561 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
Every natural number has a unique representation as a linear combination of Fibonacci numbers with binary coefficients.  ...  A third mapping, which is non- linear, is to raise the elements of the subfield to a power r.  ... 

Page 1030 of Mathematical Reviews Vol. 42, Issue 4 [page]

1971 Mathematical Reviews  
Author’s summary: “Some known results on the non- existence of linear optimal codes can be very easily proved using results on the weight distribution of such codes.” 5716 Savage, John E.  ...  IT-14 (1968), 174-176] on the minimum distance of binary arithmetic cyclic codes is given here.  ... 

Performance Comparison of Function Evaluation Methods [article]

Leo Liberti
2002 arXiv   pre-print
The outcome of the test indicates that the n-ary tree representation of algebraic expressions is the fastest method, closely followed by black-box function method, then by binary trees and lastly by string  ...  We perform a comparison of the performance and efficiency of four different function evaluation methods: black-box functions, binary trees, n-ary trees and string parsing.  ...  Some of the early work focused on how to handle floating point computation efficiently, rather than on the actual method used for the evaluation [Ash64] ; in other instances, the evaluation of certain  ... 
arXiv:cs/0206010v2 fatcat:vznnrugcwngs3djuamtxmy57yy

Page 8139 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
(English summary) Linear algebra and statistics (Istanbul, 1997). Linear Algebra Appl. 289 (1999), no. 1-3, 25-39.  ...  Summary: “Two algorithms for characterization of input error events producing specified distance at the output of certain binary- input partial-response (PR) channels are presented.  ... 

DLVM: A modern compiler infrastructure for deep learning systems [article]

Richard Wei, Lane Schwartz, Vikram Adve
2018 arXiv   pre-print
We present DLVM, a design and implementation of a compiler infrastructure with a linear algebra intermediate representation, algorithmic differentiation by adjoint code generation, domain-specific optimizations  ...  However, many of the existing deep learning frameworks are software libraries that act as an unsafe DSL in Python and a computation graph interpreter.  ...  All element-wise binary operators support broadcasting. A sample of DLVM IR code is shown in Figure 3 on the next page.  ... 
arXiv:1711.03016v5 fatcat:6rh2hyxr6zh4bikxpqf4ar6vee

Page 1380 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
Recent coding results have associated with certain binary Ker- dock codes a class of Z,-linear Kerdock codes.  ...  On the minimum distance of ternary cyclic codes; Chapter 3. Five new optimal ternary linear codes; Chapter 4. An optimal ternary [69, 5,45] code and related codes; Chapter 5.  ... 

Page 735 of Mathematical Reviews Vol. 55, Issue 2 [page]

1978 Mathematical Reviews  
for the optimal coding.”  ...  Some new bounds on the size of codes (summary).  ... 

A generative geometric kernel

Jacques Carette, Mustafa Elsheikh, Spencer Smith
2011 Proceedings of the 20th ACM SIGPLAN workshop on Partial evaluation and program manipulation - PERM '11  
The resulting code has minimal traces of the design abstractions. We achieve genericity through a layered design deriving concepts from affine geometry, linear algebra and abstract algebra.  ...  The cost of abstraction is removed by using MetaO-Caml's support for code generation coupled with some annotations atop the code type.  ...  An array is really a memory arrangement, while a matrix is a representation of a linear operator with respect to a certain basis.  ... 
doi:10.1145/1929501.1929510 dblp:conf/pepm/CaretteES11 fatcat:ec4q62jjvfdh3pzasylmgby57i

Page 6000 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
); Lahtonen, Jyrki (FIN-TURK; Turku) On algebraic decoding of the Z3-linear Calderbank-McGuire code.  ...  (One of them, the Viterbi algorithm, is based on considering quasi-cyclic codes as tailbiting convolutional codes. ) The second topic comprises a study of binary optimum burst- correcting codes.  ... 

Coding Theory (Dagstuhl Seminar 11461)

Joachim Rosenthal, M. Armin Shokrollahi, Judy L. Walker, Marc Herbstritt
2012 Dagstuhl Reports  
The purpose of this workshop is to bring together researchers in the field to discuss recent theoretical advances in algebraic coding, codes on graphs, and network coding, as well as new and emerging applications  ...  By choosing a fixed codebook, binary strings of appropriate length are injectively mapped into the elements of the code.  ...  Ring-Linear Coding started enjoying increased attention at the beginning of the last decade of the previous century, when it was discovered that certain non-linear binary block codes of high quality could  ... 
doi:10.4230/dagrep.1.11.50 dblp:journals/dagstuhl-reports/RosenthalSW11 fatcat:vsvcwodxnzc4rpd2lxi4w6zkae

Page 744 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
{For the entire collection see MR 97a:94002. } 97a:94009 94860 94B05 LeVan, Mike (1-ABRN-DS; Auburn, AL); Phelps, Kevin T. (1-ABRN-DS; Auburn, AL) Computing the kernel of a non-linear code.  ...  Summary: “This paper provides a survey of the existing literature on the decoding of algebraic-geometric codes. Definitions, theo- rems, and cross references are given.  ... 

Non-Binary Decoder Diversity for Dense or Locally-Dense Parity-Check Codes

David Declercq
2011 IEEE Transactions on Communications  
In this paper, a new and promising framework, called "non-binary decoder diversity", is presented based on the observation that different non-binary Tanner graphs of the same binary code, decoded with  ...  Index terms-non-binary Tanner graphs, non-binary belief-propagation, iterative decoding of block codes, near MLD decoders.  ...  Diversity of non-binary Tanner graph representations for the same code.  ... 
doi:10.1109/tcomm.2011.010411.080527 fatcat:ilqchmlr5rfwpiohzs5qana33q

Bridging the gap

Andreas Kunft, Alexander Alexandrov, Asterios Katsifodimos, Volker Markl
2016 Proceedings of the 3rd ACM SIGMOD Workshop on Algorithms and Systems for MapReduce and Beyond - BeyondMR '16  
Programs written in Lara are compiled to an intermediate representation (IR) which enables optimizations across linear and relational algebra.  ...  Second, it aims to enable joint optimizations over both relational and linear algebra.  ...  Listing 1 : 1 Code snippet written in Lara representation for joint optimization of linear and relational algebra.  ... 
doi:10.1145/2926534.2926540 dblp:conf/sigmod/KunftAKM16 fatcat:4zusweokhbda5hxc64i6hr4nb4
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