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The smallest length of binary 7–dimensional linear codes with prescribed minimum distance

van Tilborg Henk C.A.
1981 Discrete Mathematics  
Le; n(k, d) denote the smallest value of n for which a binary (n, k, d) code exists. Then nik, d) ~51s known ior all d, when k 66. All values of n(7, d) will now be presented.  ...  In [7] one can find a table of values of resp. bounds on -the minimum distance d of an (n, k, &code i.e. a binary, linear, k-dimensional code of length n with minimum distance d.  ...  Let G be the generator :natric of a binary linear code C with top row c.  ... 
doi:10.1016/0012-365x(81)90166-7 fatcat:dzcrlfnlizhkdjjhiu6y7b2hfm

Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance [chapter]

Axel Kohnert, Sascha Kurz
2008 Lecture Notes in Computer Science  
In this paper we construct constant dimension space codes with prescribed minimum distance.  ...  We will modify a method of Braun, Kerber and Laue which they used for the construction of designs over finite fields to do the construction of space codes.  ...  Taking the union of the corresponding 16 orbits on the 3-spaces of GF (2) 7 we get a constant dimension code with 304 codewords having minimum distance 4.  ... 
doi:10.1007/978-3-540-89994-5_4 fatcat:rw75xnaoabfv5pi4twlfdvnlou

Adding a parity-check bit

J. Simonis
2000 IEEE Transactions on Information Theory  
The smallest length of binary 7-dimensional linear codes with prescribed minimum distance," Discr. Math., vol. 33, pp. 197-207, 1981. [38] S.  ...  and minimum distance to be extendable to a code of the same dimension, length + 1, and minimum distance Index Terms-Code extension, linear codes, parity-check bit.  ...  C, where G and C are finite groups with C Abelian, which satisfy a particular quasi-associative equation (1) . They arise naturally in the topology of surfaces, in  ... 
doi:10.1109/18.850691 fatcat:bndeo5zrbzgqdp4n56nzf6zsku

Author index

1981 Discrete Mathematics  
., The smallest length of binary 7-dimensional linear codes with prescribed minimum distance Walther, H.. Note on two problems of J.  ...  ., Long unimod31 su~sequcnces: a problem of F.R.K. Chung (Note) Stinson, D.R., A general construction for group-divisible designs Straight, H.J., see J.F. Fink Thomason, A.G., see E.J.  ... 
doi:10.1016/0012-365x(81)90278-8 fatcat:34ez72pakzclxad2o7kykqcei4

Page 5224 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
Some new results on the function t{n, k] (the smallest covering radius of any binary linear code with length n and dimension k) are given: t(38,6] > 13, #[47,7] > 16, t{59, 8] > 20.” 91i1:94047 94B75 Solé  ...  We also present an upper bound on the dimension of any linear code over GF(q) of length n, minimum Hamming distance d, and contraction index A.  ... 

Classification of 8-divisible binary linear codes with minimum distance 24 [article]

Sascha Kurz
2020 arXiv   pre-print
We classify 8-divisible binary linear codes with minimum distance 24 and small length. As an application we consider the codes associated to nodal sextics with 65 ordinary double points.  ...  The minimum distance d of a linear code is the smallest non-zero weight of a codeword.  ...  Here we study the special case of triply even, i.e., 8-divisible binary linear codes with minimum distance d = 24. We exhaustively enumerate all such codes for small lengths.  ... 
arXiv:2012.06163v1 fatcat:x3klgnhnsrfnzbhdxobkonbg6a

Page 440 of Mathematical Reviews Vol. , Issue 82a [page]

1982 Mathematical Reviews  
A. 82a:94066 The smallest length of binary 7-dimensional linear codes with prescribed minimum distance. Discrete Math. 33 (1981), no. 2, 197—207.  ...  Author’s summary: “Let n(k,d) denote the smallest value of n for which a binary (,k,d) code exists. Then n(k,d) was known for all d, when k < 6. All values of n(7,d) are presented.”  ... 

Self-Orthogonal Codes Constructed From Posets and Their Applications in Quantum Communication

Yansheng Wu, Yoonjin Lee
2020 Mathematics  
Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7.  ...  Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the  ...  Acknowledgments: We thank the reviewers of this paper for their helpful comments, which improved the clarity of this paper. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/math8091495 fatcat:tgrwhzrgpvdtph7esmcqricapy

A projective two-weight code related to the simple group Co_1 of Conway [article]

Bernardo G. Rodrigues
2018 arXiv   pre-print
Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of Co_1.  ...  A binary [98280, 24, 47104]_2 projective two-weight code related to the sporadic simple group Co_1 of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module  ...  Moreover, a code with minimum distance d = 3 and covering radius 2 is called uniformly packed if every vector which is not a codeword is at distance 1 or 2 from a constant number of codewords [24, Theorem  ... 
arXiv:1804.00283v1 fatcat:42evmztnmfcqdjf43p3iquhzxu

PIR Codes with Short Block Length

Sascha Kurz, Eitan Yaakobi
2021 Designs, Codes and Cryptography  
The main problem under this paradigm is to minimize the number of encoded bits given the values of s and k, where this value is denoted by P(s, k) .  ...  The main focus of this work is to analyze P(s, k) for a large range of parameters of s and k. In particular, we improve upon several of the existing results on this value.  ...  A binary linear code of length n and dimension s will be denoted by [n, s] or [n, s, d], where d denotes its minimum Hamming distance. The set [n] denotes the set of integers {1, 2, . . . , n}.  ... 
doi:10.1007/s10623-020-00828-6 fatcat:53uc6oy66fearfst5u5ydis7zy

PIR Codes with Short Block Length [article]

Sascha Kurz, Eitan Yaakobi
2020 arXiv   pre-print
The main problem under this paradigm is to minimize the number of encoded bits given the values of s and k, where this value is denoted by P(s,k).  ...  The main focus of this work is to analyze P(s,k) for a large range of parameters of s and k. In particular, we improve upon several of the existing results on this value.  ...  By d ⊥ we denote the minimum distance of C ⊥ , which is also called the dual minimum distance. Lemma 10. Let C be a linear [n, s, d] code with minimum dual distance d ⊥ and generator matrix G.  ... 
arXiv:2001.03433v1 fatcat:z5zsa4cyt5f7noploeop6tqd34

Constructions and properties of block codes for partial-response channels

L.M.G.M. Tolhuizen, K.A. Schouhamer Immink, H.D.L. Hollmann
1995 IEEE Transactions on Information Theory  
Han for pointing out the game-theoretic aspect of the competitively optimal coding, and Prof. T. M. Cover and Prof. K. Itoh for their helpful comments.  ...  If we use a binary code C of length m and minimum distance d, we obtain a code for the 1 -D channel of length 2m with I C I words and d,& > 1-f 2d.  ...  CODES WITH d~i, 2 2 By application of the construction methods given in the previous sections, codes can be designed with prescribed length and minimum distance.  ... 
doi:10.1109/18.476329 fatcat:vt4ytixkozhqfj7ppvipnrzhlm

Conservative arrays: multidimensional modulation codes for holographic recording

A. Vardy, M. Blaum, P.H. Siegel, G.T. Sincerbox
1996 IEEE Transactions on Information Theory  
dimension is determined by the minimum distance of the corresponding code. sional modulation constraints, error-correcting codes.  ...  Using n binary c o d e w n e per dimension-with minimum Hamming distance d 2 2t -3, we apply a certain transformation to an arbitrary information array which ensures that the number of transitions in each  ...  Using differential coding, the pigeonhole principle (cf. [7]), and k binary codes-one per dimension-with minimum Hamming distance d 2 2t -3, we derive an efficient algorithm for encoding unconstrained  ... 
doi:10.1109/18.481792 fatcat:2zplbrqbmzftbd4ddygjmfxpjq

Error Correction for Index Coding With Side Information

Son Hoang Dau, Vitaly Skachek, Yeow Meng Chee
2013 IEEE Transactions on Information Theory  
This question turns out to be a generalization of the problem of finding a shortest length error-correcting code with a prescribed error-correcting capability in the classical coding theory.  ...  For large alphabets, a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the Singleton bound.  ...  ACKNOWLEDGMENT The authors would like to thank the authors of [7] for providing a preprint of their paper.  ... 
doi:10.1109/tit.2012.2227674 fatcat:uaaat43sdrh7vexvho5wbmyo6u

Trellis Coded Block Codes: Design and Applications

Ganesan Thiagarajan, Chandra R Murthy
2012 Journal of Communications  
The proposed uniform sub-set partitioning is used to increase the minimum distance of the code, as in trellis coded modulation (TCM).  ...  The TCB code (TCBC) construction is based on an algebraic structure inherent to many LBCs, which allows one to partition an LBC into sub-sets with a constant distance between every pair of code words in  ...  The work of the first author was supported in part by Texas Instruments India Pvt. Ltd.  ... 
doi:10.4304/jcm.7.1.73-85 fatcat:q3ipccyiufge5lgpdqvy5uiuoy
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