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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  ...  Some nontrivial properties of perfect binary codes are discussed.  ...  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

A generalized concatenation construction for q-ary 1-perfect codes [article]

Alexander M. Romanov
2017 arXiv   pre-print
In this paper, a generalized concatenation construction for q-ary 1-perfect codes is presented that allows us to construct q-ary 1-perfect codes of length (q - 1)nm + n + m from the given q-ary 1-perfect  ...  This construction allows us to also construct q-ary codes with parameters (q^s_1 + s_2, q^q^s_1 + s_2 - (s_1 + s_2) - 1, 3)_q and can be regarded as a q-ary analogue of the well-known Phelps construction  ...  In the binary case, the concatenation construction is based on partitions of two types -a partition of the space F n 2 into 1-perfect codes of length n and a partition of the binary MDS code of length  ... 
arXiv:1711.00189v1 fatcat:5xteig2s55ckpal4najhtkur4q

On non-full-rank perfect codes over finite fields [article]

Alexander M. Romanov
2017 arXiv   pre-print
We suggest a generalization of the concatenation construction to the q-ary case and construct the ternary 1-perfect codes of length 13 and rank 12.  ...  We derive necessary and sufficient conditions for q-ary 1-perfect codes of non-full rank.  ...  construction constructed a binary 1-perfect code of length 15, which is not the code of Vasil'ev [3] .  ... 
arXiv:1704.02627v1 fatcat:fjebxtb7ujetjghqcu7czj577m

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  ...  Propelinear full rank perfect codes of lengths 15 and 31 Let us recall the Vasil'ev construction [14] . Let C be a perfect binary code of length (n−1)/2.  ...  Recall the Mollard construction for binary codes. Let C t and C m be any two perfect codes of lengths t and m, respectively, containing all-zero vectors.  ... 
arXiv:1210.8253v1 fatcat:flowe3q6ivcu3agfn3q67bmzzy

On $Z_{2^k}$-Dual Binary Codes

Denis S. Krotov
2007 IEEE Transactions on Information Theory  
The new generalization Φ: Z_2^k^n → Z_2^2^k-1n is connected with the known generalized Gray map ϕ in the following way: if we take two dual linear Z_2^k-codes and construct binary codes from them using  ...  the generalizations ϕ and Φ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity.  ...  We first introduce the concept of 1-perfect code, which is a generalization of the concept of extended 1-perfect binary code to some nonbinary cases.  ... 
doi:10.1109/tit.2007.892787 fatcat:mmndf2psijc7xovgwrlr5h6o2q

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.  ...  Since there is oneto-one correspondence between 1-perfect binary codes and extended 1-perfect binary codes, the results are formulated in terms of extended 1-perfect binary codes. II.  ...  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.  ... 
doi:10.1109/tit.2008.917692 fatcat:t34bhbajofavfanqe7efxupws4

Perfect binary codes: constructions, properties, and enumeration

T. Etzion, A. Vardy
1994 IEEE Transactions on Information Theory  
Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties.  ...  Moreover, for any m 2 4 we construct jidl-rank perfect binary codes of length 2" -1. These codes are obviously nonequivalent to any of the previously known perfect codes.  ...  a perfect binary code of full rank.  ... 
doi:10.1109/18.335887 fatcat:2a67nll72bh6tjpwzeu2fu37cq

On the structure of non-full-rank perfect $q$-ary codes

Denis Krotov, Olof Heden
2011 Advances in Mathematics of Communications  
The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by  ...  The Phelps general product construction of perfect binary code from 1984 is generalized to obtain μ-components, and new lower bounds on the number of perfect 1-error-correcting q-ary codes are presented  ...  In [3] it was shown that a binary perfect code can be constructed as the union of different subcodes (μcomponents) satisfying some generalized parity-check property, each of them being constructed independently  ... 
doi:10.3934/amc.2011.5.149 fatcat:mafi3tvyxng6hjpgplk4bins6u

Page 5064 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
Summary: “Properties of nonlinear perfect binary codes are inves- tigated and several new constructions of perfect codes are derived from these properties.  ...  Moreover, for any m > 4 we construct full-rank perfect binary codes of length 2” — 1. These codes are obviously nonequivalent to any of the previously known perfect codes.  ... 

On full-rank perfect codes over finite fields

A. M. Romanov
2016 Journal of Applied and Industrial Mathematics  
This construction is a generalization of the Etzion and Vardy construction of full-rank binary 1-perfect codes (1994).  ...  The switching construction of 1-perfect codes are generalized for the q-ary case.  ...  Next, we describe so called doubling construction of binary 1-perfect codes (or extended binary 1-perfect codes).  ... 
doi:10.1134/s1990478916030157 fatcat:zlxzwuri2veghmoxj5yec2fhze

Two optimal one-error-correcting codes of length 13 that are not doubly shortened perfect codes

Patric R. J. Östergård, Olli Pottonen
2010 Designs, Codes and Cryptography  
Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such (13,512,3) codes.  ...  The doubly shortened perfect codes of length 13 are classified utilizing the classification of perfect codes in [P.R.J. Östergård and O.  ...  Can the structure of the two codes be generalized or is there some construction that can be applied to them to obtain an infinite family of similar codes?  ... 
doi:10.1007/s10623-010-9450-4 fatcat:tnkrlcmwkjeslnbopsufikpmmm

Sub-Classifier Construction for Error Correcting Output Code Using Minimum Weight Perfect Matching [article]

Patoomsiri Songsiri, Thimaporn Phetkaew, Ryutaro Ichise, Boonserm Kijsirikul
2013 arXiv   pre-print
We propose a method for constructing the Error Correcting Output Code to obtain the suitable combination of positive and negative classes encoded to represent binary classifiers.  ...  The minimum weight perfect matching algorithm is applied to find the optimal pairs of subset of classes by using the generalization performance as a weighting criterion.  ...  Some binary classifiers include the large number of classes that seem to be complex to construct binary models with high accuracy (as the first important issue).  ... 
arXiv:1312.7179v1 fatcat:jjpsbuakrvgvvjahgzedbokpsm

Editorial: 3rd International Castle Meeting on Coding Theory and Applications

Joaquim Borges, Mercè Villanueva, Victor Zinoviev
2012 Designs, Codes and Cryptography  
Some works investigate Constructions of some special classes of codes.  ...  codes and binary abelian codes, based on the computation of Groebner basis.  ... 
doi:10.1007/s10623-012-9775-2 fatcat:pa276hqqgja4fjsq44biyataja

Generalized perfect arrays and menon difference sets

Jonathan Jedwab
1992 Designs, Codes and Cryptography  
The array is perfect ifz = (0, .... 0) and binary if the array elements are all +1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.  ...  Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence.  ...  We illustrate the recursive application of the construction theorems to the binary case in Section 9, where we obtain the main constructive and nonexistence results of the paper.  ... 
doi:10.1007/bf00124209 fatcat:6cxiheaiwjgdlk5scneq6oy3mm

Page 9004 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
Since perfect codes of length n = 3 and 7 are unique, the existence of nonsystematic perfect binary codes is fully established.  ...  Finally, nonsystematic perfect codes of length 15 and 31 are found by computer construction.  ... 
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