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On Bounds for Ring-Based Coding Theory [article]

Niklas Gassner, Marcus Greferath, Joachim Rosenthal, Violetta Weger
2021 arXiv   pre-print
A further highlight is the proof of a Johnson bound for the homogeneous weight on a general finite Frobenius ring.  ...  coding theory over finite fields.  ...  Let us recall some preliminaries in coding theory, where we focus on ring-linear coding theory. For q a prime power, let us denote by F q the finite field with q elements.  ... 
arXiv:2103.07749v2 fatcat:zzfgazxxzjd37gztoeggo3ayne

Coding Theory (Dagstuhl Seminar 11461)

Joachim Rosenthal, M. Armin Shokrollahi, Judy L. Walker, Marc Herbstritt
2012 Dagstuhl Reports  
Coding theory is all about the design of excellent codes as a function of the communications channel, and the design of efficient algorithms for choosing the codebook vectors, and more importantly, for  ...  As such, successful design of codes requires knowledge and tools in a number of areas such as combinatorics, algorithms design, probability theory and complexity theory, to name a few.  ...  (for decoding, list-decoding, and Gröbner bases computation); elliptic curve methods and bent and hyperbent functions; rank-metric codes; bounds on codes (semidefinite programming bounds, BCH-like bounds  ... 
doi:10.4230/dagrep.1.11.50 dblp:journals/dagstuhl-reports/RosenthalSW11 fatcat:vsvcwodxnzc4rpd2lxi4w6zkae

Page 4301 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
4301 94B Theory of error-correcting codes and error-detecting codes 2004e:94037 94A62 Chen, Xiao Ming (PRC-ASBJ-S; Beijing) An algorithm for deciding module-matroid over Galois rings. (Chinese.  ...  The authors develop a unified theory of Grébner bases over a principal ideal ring with a view to applications in coding, e.g., to describing the structure of codes over Galois rings or, more generally,  ... 

Page 3149 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
Theory 48 (2002), no. 1, 112-122. Low density parity-check (LDPC) codes were introduced by Gal- lager in 1963. These codes are based on an ensemble of bipartite graphs.  ...  Summary: “In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of alge- braic geometric error-correcting codes constructed from varieties of arbitrary  ... 

The Art of DNA Strings: Sixteen Years of DNA Coding Theory [article]

Dixita Limbachiya and Bansari Rao and Manish K. Gupta
2016 arXiv   pre-print
Thus it is natural to expect that coding theory will be the foundational subject for the DNA computing paradigm.  ...  The classical coding theory techniques have provided foundation for the current information and communication technology (ICT).  ...  With emerging area of algebraic coding and biological coding theory, these challenges can be investigated and resolved. 11) Bounds on reverse code for d = 3 -This bound is computed from the concept  ... 
arXiv:1607.00266v1 fatcat:qboafglvkbay7jw2lnycqsxs2u

Page 9006 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
Theory 45 (1999), no. 2, 701-702. Summary: “In 1976, Moore investigated a class of self-dual double-circulant codes based on primes of the form q = 3 mod 8.  ...  (RS-AOS-M; Moscow) New lower bounds on aperiodic crosscorrelation of binary codes. (English summary) IEEE Trans. Inform. Theory 45 (1999), no. 1, 284-288.  ... 

Page 3405 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
In Part II, trellis codes based on group systems are considered as a special case for which code and encoder constructions are particularly simple.  ...  (IL-TLAV-EN; Tel Aviv) Linear programming bounds for doubly-even self-dual codes. (English summary) IEEE Trans. Inform. Theory 43 (1997), no. 4, 1238-1244.  ... 

Page 9085 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
The representation method based on the lattice theory, that we call the lattice PWL model, is a form that fits that scheme.  ...  When A = W,(k), an estimate for the minimum Eu- clidean weight of the trace code is derived from a bound on an exponential sum.  ... 

Page 5221 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
It is shown that a method by Edel to construct codes which improve the bounds on optimal linear codes can be obtained by this approach. T.  ...  The author’s new approach to obtaining this distribution is based on the notion of an objective function.  ... 

Page 7269 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
Constructions of (2,2)-separating systems derived from error- correcting codes are given, together with bounds on their pa- rameters based on distance considerations.”  ...  The E(R) is positive below a threshold Ro, which implies Rg is a lower bound on the quantum capacity. The proof of the main result is based on the method of types. Mariam E.  ... 

Page 7633 of Mathematical Reviews Vol. , Issue 2001J [page]

2001 Mathematical Reviews  
Brouwer, “Bounds on the minimum distance of lin- ear codes”,] based on our codes.” 2001j:94052 94B60 11H71 Bachoc, Christine (F-BORD-ALG; Talence) ; Gulliver,  ...  Theory 46 (2000), no. 7, 2638-2642. Summary: “Based on a construction of C. P Xing, H. Niederreiter and K.-Y. Lam [IEEE Trans. Inform.  ... 

Page 7265 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
The code construction is based on the factorization of x* — 1 over the subgroup of units of an extension ring.  ...  Theory 43 (1997), no. 6, 2023-2026] sufficient conditions for a linear block code to be good or proper for error detection.  ... 

Page 2511 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
ON) Trellis-based decoding of binary linear block codes.  ...  State-extension and branch- labeling is based only on the state labels at the current position and on the set of atomic generators, making it unnecessary to store the whole code tree.  ... 

Cumulative subject index volumes 56–59

1983 Information and Control  
representations of graphs, 56, 183 Synchronization trees rigid, and concurrency theory connections, 57, 102 Synchronous experiments on processes, 59, 36 T Target code production, with explicit  ...  , 59, 1 propositional temporal and extended temporal: axiomatization, decision procedure, and complexity, 56, 72 Loops in combinator based compilers, 57, 148 Lower bound on monotone network complexity  ... 
doi:10.1016/s0019-9958(83)80037-0 fatcat:dtwnjwpfbndyfc26jagdqcnmwa

Page 1504 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
This definition was motivated by two recent trends in coding theory: the study of algebraic geometric codes over finite fields, and the study of codes over rings.  ...  make decisions based on the correlation be- tween pseudosignals and the channel output.  ... 
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