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

1981
*
Discrete Mathematics
*

Le; n(k, d) denote

doi:10.1016/0012-365x(81)90166-7
fatcat:dzcrlfnlizhkdjjhiu6y7b2hfm
*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. ...##
###
Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance
[chapter]

2008
*
Lecture Notes in Computer Science
*

In this paper we construct constant dimension space

doi:10.1007/978-3-540-89994-5_4
fatcat:rw75xnaoabfv5pi4twlfdvnlou
*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. ...##
###
Adding a parity-check bit

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 ...

##
###
Author index

1981
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(81)90278-8
fatcat:34ez72pakzclxad2o7kykqcei4
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

We classify 8-divisible

arXiv:2012.06163v1
fatcat:x3klgnhnsrfnzbhdxobkonbg6a
*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*. ...##
###
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

2020
*
Mathematics
*

Finally, we obtain four infinite families

doi:10.3390/math8091495
fatcat:tgrwhzrgpvdtph7esmcqricapy
*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. ...##
###
A projective two-weight code related to the simple group Co_1 of Conway
[article]

2018
*
arXiv
*
pre-print

Similarly,

arXiv:1804.00283v1
fatcat:42evmztnmfcqdjf43p3iquhzxu
*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 ...##
###
PIR Codes with Short Block Length

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}. ...

##
###
PIR Codes with Short Block Length
[article]

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. ...

##
###
Constructions and properties of block codes for partial-response channels

1995
*
IEEE Transactions on Information Theory
*

Han for pointing out

doi:10.1109/18.476329
fatcat:vt4ytixkozhqfj7ppvipnrzhlm
*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*. ...##
###
Conservative arrays: multidimensional modulation codes for holographic recording

1996
*
IEEE Transactions on Information Theory
*

dimension is determined by

doi:10.1109/18.481792
fatcat:2zplbrqbmzftbd4ddygjmfxpjq
*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 ...##
###
Error Correction for Index Coding With Side Information

2013
*
IEEE Transactions on Information Theory
*

This question turns out to be a generalization

doi:10.1109/tit.2012.2227674
fatcat:uaaat43sdrh7vexvho5wbmyo6u
*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. ...##
###
Trellis Coded Block Codes: Design and Applications

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. ...

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