Filters








14,238 Hits in 5.7 sec

Rank and Kernel of Binary Hadamard Codes

K.T. Phelps, J. Rifa, M. Villanueva
2005 IEEE Transactions on Information Theory  
Lower and upper bounds for the rank and dimension of the kernel of a Hadamard code of length n = 2 t , are also established.  ...  Finally, we construct Hadamard codes for all possible ranks and dimension of kernels between these bounds.  ...  BOUNDS ON THE RANK AND THE DIMENSION OF THE KERNEL In this section we will give an upper and lower bound on the rank, in terms of the dimension of the kernel.  ... 
doi:10.1109/tit.2005.856940 fatcat:2vrvexbejnbsfg4cmcu3czag4e

Page 7267 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
In this paper, the authors establish upper and lower bounds on the rank and dimension of the kernel of perfect binary codes. Moreover, using the doubling and switching construction [K. T.  ...  Summary: “The rank and kernel of l-perfect additive codes is determined.  ... 

On rank and kernel of some mixed perfect codes

Fabio Pasticci, Thomas Westerbäck
2009 Discrete Mathematics  
A lower and an upper bound for the rank k of the kernel of mixed perfect 1-error correcting codes in Z (n, l), depending on the rank r of the mixed perfect code and the structure of the corresponding dual  ...  Due to a general construction of mixed perfect 1-error correcting group codes in Z (n, l), we show that the upper bound is tight for some n, l and r.  ...  Acknowledgements This paper is a result of a research collaboration in connection with a two-week course on perfect codes given by Olof Heden in June 2005 at the University of Perugia, Italy.  ... 
doi:10.1016/j.disc.2008.06.037 fatcat:nmm2znetzba5jllz5mp4ywft2q

Ranks of propelinear perfect binary codes [article]

George K. Guskov, Ivan Yu. Mogilnykh, Faina I. Solov'eva
2012 arXiv   pre-print
It is proven that for any numbers n=2^m-1, m >= 4 and r, such that n - log(n+1)<= r <= n excluding n = r = 63, n = 127, r in 126,127 and n = r = 2047 there exists a propelinear perfect binary code of length  ...  n and rank r.  ...  The authors cordially thank Fedor Dudkin for useful discussions.  ... 
arXiv:1210.8253v1 fatcat:flowe3q6ivcu3agfn3q67bmzzy

Z2Z4-linear codes: rank and kernel [article]

Cristina Fernandez-Cordoba, Jaume Pujol, Merce Villanueva
2009 arXiv   pre-print
Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z2Z4-additive code for each possible pair (r,k) is given.  ...  In this paper, the rank and dimension of the kernel for Z2Z4-linear codes, which are the corresponding binary codes of Z2Z4-additive codes, are studied.  ...  PAIRS OF RANK AND KERNEL DIMENSION OF Z 2 Z 4 -ADDITIVE CODES In this section, once the dimension of the kernel is fixed, lower and upper bounds on the rank are established.  ... 
arXiv:0807.4247v2 fatcat:jbfesqrawjeyjbbhy4l2zgriaq

Perfect binary codes: bounds and properties

F.I. Solov'eva
2000 Discrete Mathematics  
We consider some constructions of perfect binary codes with the purpose to outline bounds on the number of nonequivalent perfect binary codes and we present the best known lower and upper bounds on the  ...  number of di erent perfect binary codes.  ...  Acknowledgements The author is grateful to Werner Heise and the anonymous referees for comments improving the presentation of the paper.  ... 
doi:10.1016/s0012-365x(99)00188-0 fatcat:u7ndxiyxlff65cpjlrwfhh3xam

On the classification of perfect codes: Extended side class structures

Olof Heden, Martin Hessler, Thomas Westerbäck
2010 Discrete Mathematics  
class an invariant L C and this invariant was shown to be a subspace of the kernel of some perfect code.  ...  The two 1-error correcting perfect binary codes, C and C are said to be equivalent if there exists a permutation π of the set of the n coordinate positions and a wordd such that C = π (d + C ).  ...  One typical example is for perfect codes of length 31, rank 27 and kernel of dimension 24, the number of linear equivalence classes is one, while the number of equivalence classes is 197; see [8, 9] .  ... 
doi:10.1016/j.disc.2009.07.023 fatcat:767jnbrrxvfilffemlgoimezmy

A concatenation construction for propelinear perfect codes from regular subgroups of GA(r,2) [article]

I.Yu.Mogilnykh, F. I. Solov'eva
2019 arXiv   pre-print
In the paper new propelinear perfect binary codes of any admissible length more than 7 are obtained by a particular case of the Solov'eva concatenation construction–1981 and the regular subgroups of the  ...  A code C is called propelinear if there is a subgroup of Aut(C) of order |C| acting transitively on the codewords of C.  ...  The kernel problem as far as the rank and kernel problem is still open for propelinear perfect binary codes.  ... 
arXiv:1905.10005v2 fatcat:fqfjl24kund6zh7r7dboiwirdq

A concatenation construction for propelinear perfect codes from regular subgroups of GA (r,2)

I. Yu. Mogilnykh, F. I. Solov'eva
2019 Sibirskie Elektronnye Matematicheskie Izvestiya  
In the paper new propelinear perfect binary codes of any admissible length more than 7 are obtained by a particular case of the Solov'eva concatenation construction-1981 and the regular subgroups of the  ...  A code C is called propelinear if there is a subgroup of Aut(C) of order |C| acting transitively on the codewords of C.  ...  rank and kernel and, in particular, codes of prefull rank n − 1 and codes with the dimension of the kernel n − 2 log n − 2.  ... 
doi:10.33048/semi.2019.16.119 fatcat:krn4uawmbzg4zgwqvicsqx7v44

The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties

Patric R. J. Ostergard, Olli Pottonen, Kevin T. Phelps
2010 IEEE Transactions on Information Theory  
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. Östergård and O.  ...  Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.  ...  ACKNOWLEDGMENTS The authors thank Olof Heden, Dmitrii Zinov'ev, and Victor Zinov'ev for helpful discussions.  ... 
doi:10.1109/tit.2010.2046197 fatcat:p7zakjehxvbudep5cxppwo4dka

On the Number of $1$-Perfect Binary Codes: A Lower Bound

Denis S. Krotov, Sergey V. Avgustinovich
2008 IEEE Transactions on Information Theory  
We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.  ...  The construction of 1-perfect binary codes presented in the current paper gives the most powerful known class of such codes and leads to a lower bound on their number.  ...  A LOWER BOUND ON THE NUMBER OF 1-PERFECT CODES Denote by e KLA(n) the number of different extended 1-perfect codes given by Construction 3.  ... 
doi:10.1109/tit.2008.917692 fatcat:t34bhbajofavfanqe7efxupws4

Rank and Kernel of F_p-Additive Generalised Hadamard Codes [article]

Steven T. Dougherty, Josep Rifà, Mercè Villanueva
2020 arXiv   pre-print
Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are F_p-additive are established.  ...  For specific ranks and dimensions of the kernel within these bounds, F_p-additive GH codes are constructed.  ...  Bounds on the rank and dimension of the kernel In this section, we state new results on the rank and dimension of the kernel for F p -additive generalised Hadamard codes.  ... 
arXiv:2001.11609v1 fatcat:wmw55lsvifdjhig2zma2l7bxou

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

Mercè Villanueva, Fanxuan Zeng, Jaume Pujol
2014 Designs, Codes and Cryptography  
Moreover, some properties and constructions of new codes from given ones in terms of this representation are described.  ...  Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes.  ...  Proposition 3 [13] Let C be a binary code of length n with rank , dimension of the kernel κ and parity-check system (H S). Then, = n − rank(H) + rank(S) and κ = n − rank(H).  ... 
doi:10.1007/s10623-014-0028-4 fatcat:3gouxcncfrhzna6eyh4qnbvd74

Ranks and Kernels of Codes From Generalized Hadamard Matrices

Steven T. Dougherty, Josep Rifa, Merce Villanueva
2016 IEEE Transactions on Information Theory  
For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.  ...  Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes.  ...  Moreover, for some particular cases, we specify lower and upper bounds on the rank, once the dimension of the kernel is given.  ... 
doi:10.1109/tit.2015.2509061 fatcat:uiv5dqtusjdlfmjfeccjw3bg4a

Page 1542 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
[Heden, Olof] (S-RIT; Stockholm) Perfect codes of complete rank with kernels of large dimensions. (Russian. Russian summary) Diskretn. Anal. Issled. Oper. Ser. 1 8 (2001), no. 4, 3-8.  ...  ° — 1 of complete rank with kernels of all possible dimensions K from (n — 1)/2 to U(n), which is the maximum possible.  ... 
« Previous Showing results 1 — 15 out of 14,238 results