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New bounds for the minimum length of quaternary linear codes of dimension five

Iliya G. Boukliev
1997 Discrete Mathematics  
Let n4(k,d) be the smallest integer n, such that a quaternary linear [n,k,d]-code exists.  ...  The bounds n4(5, 21 ) ~< 32, n4(5, 30) = 43, n4(5, 32) = 46, n4(5, 36) = 51, n4(5, 40) ~< 57, n4(5, 48 ) ~ 67, /'/4(5, 64) = 88, n4(5, 68)~<94, n4(5, 70)~<97, n4(5, 92)~< 126, n4(5, 98)~<135, n4(5, 122  ...  Introduction and preliminary results Let GF(q) denote the Galois field of q elements. A quaternary linear code of length n, dimension k, and minimum Hamming distance d is called an [n, k, d]-code.  ... 
doi:10.1016/s0012-365x(96)00104-5 fatcat:ypzqqtzfo5fnhdapixxnidoidi

Page 1345 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
A survey of the results of recent work on bounds for quaternary lin- ear codes in dimensions four and five is made and a table with lower and upper bounds for d4(n, 5) is presented.” 98b:94018 94B05 11T71  ...  Ivan Landgev (Sofia) 98b:94021 94B05 51E22 Maruta, Tatsuya On the minimum length of g-ary linear codes of dimension five. (English summary) Geom. Dedicata 65 (1997), no. 3, 299-304.  ... 

Generalized Hamming weights of nonlinear codes and the relation to the Z/sub 4/-linear representation

I. Reuven, Y. Be'ery
1999 IEEE Transactions on Information Theory  
The new definition is invariant under translates of the code, it satisfies the property of strict monotonicity and the generalized Singleton bound.  ...  These generalized weights are determined by the entropy/length profile of the code. We show that this definition characterizes the performance of nonlinear codes on the wire-tap channel of type II.  ...  The discussions with him while the second author visited ENST, Paris, were the origin of the new definition of the GHW. The authors also wish to thank Y. Shany for helpful discussions.  ... 
doi:10.1109/18.749020 fatcat:c6ots4qjznarxh3et2cdhuwvmq

Author index to volume 202 (1999)

1999 Discrete Mathematics  
Sun, On independent domination number of regular graphs (l-3) 135-144 Landjev, I.N. and T. Maruta, On the minimum length of quaternary linear codes of dimension five Leng, G. and X.  ...  Li, A new upper bound on the number of edges in a geodetic graph Maruta, T., see I.N. Landjev Peterson, D. and D.R.  ... 
doi:10.1016/s0012-365x(99)00128-4 fatcat:k3esaislcrgnbghn4rzbwo2xmu

Page 1668 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
For r>2, m=m, =2' —1, let Q,, be the m-cube, 2, be the Hamming shell of the linear Hamming code of length m in Q,,, and, for 1 < j <™m, let e}, be the standard binary m-tuple basis element with a | in  ...  A 25 (1978), no. 3, 288-318; MR 80f:94016] that a self-dual quaternary code of length n has minimum weight d < 2|n/6| + 2.  ... 

On extremal additive $\mathbb {F}_4$ codes of length $10$ to $18$

Christine Bachoc, Philippe Gaborit
2000 Journal de Théorie des Nombres de Bordeaux  
The 5 extremal Type II codes of length 14 with the highest automorphism group orders among the 490 found (respectively 28,36,48,84 and 6552).  ...  Matrices for the five new codes which are characterized by the number of their automorphisms (respectively 18, 28, 36, 48, 84) are given in the Appendix.  ...  Two of these codes (QC-10r and QC-10s) are linear and were already classified as F4 linear self-dual hermitian codes in [9] and five non-linear codes (QC-10a, b, c, d, e) were found in [6] , the 12  ... 
doi:10.5802/jtnb.278 fatcat:ixrwxizvlfcpfgvhbvj3puaewa

Some ternary and quaternary codes and associated sphere packings

F.R. Kschischang, S. Pasupathy
1992 IEEE Transactions on Information Theory  
+ U I U ) construction, a new ternary code construction technique that can be used to construct the ternary Reed-Muller codes (and others), and 4) tables of linear ternary and quaternary codes obtained  ...  Results include 1) tables of the best ternary and quaternary constacyclic codes (including cyclic codes) up to block length 50, 2 ) a class of optimal [ n , 21 codes over GF(q), 3) the ( U + U + w I2u  ...  TABLES OF TERNARY AND QUATERNARY LINEAR CODES Tables V and VI list the parameters of various ternary and quaternary linear codes.  ... 
doi:10.1109/18.119683 fatcat:ldqhs3fuanchfeac2mbsvct3ri

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound [article]

Kenneth W. Shum, Jie Hao
2021 arXiv   pre-print
We classify all quaternary locally repairable codes that attain the Singleton-type upper bound for minimum distance.  ...  For each combination of achievable code parameters, an explicit code construction is given.  ...  For dimension 4, the largest code length is 17, and for dimension 3, the largest code length is 16. For dimension k = 4, the longest code with defect 2 is an ovoid code.  ... 
arXiv:2112.05377v1 fatcat:zhj4abruvvazdgdgzprtpnfkli

Page 9502 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
known bounds on the minimum weight for linear codes over Fs; are constructed from conference matrices.  ...  Summary: “We say that a binary code of length n is additive if it is isomorphic to a subgroup of Z5 x Zi , where the quaternary For the web version of Mathematical Reviews, see http: //www.ams.org/mathscinet  ... 

Detection of connectivity for regions represented by linear quadtrees

Irene Gargantini
1982 Computers and Mathematics with Applications  
The data structure used to represent the region (or image) is the linear quadtree, a recently introduced structure which basically consists of a sorted array of quaternary integers.  ...  An algorithm is given for the determination of all connected components of a given region.  ...  input: the sorted array of quaternary codes, LINQUAD, and its length. output: sets of quaternary codes, identified by BUCKET. Each set represents a connected component.  ... 
doi:10.1016/0898-1221(82)90014-1 fatcat:liztrnd6tzhwhnf6imtweds72m

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Ching-Yi Lai, Alexei Ashikhmin
2018 IEEE Transactions on Information Theory  
On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of small quantum codes  ...  As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.  ...  ACKNOWLEDGMENT We thank Todd Brun for helpful discussion. We are thankful to Markus Grassl for comments and suggestions on a previous version of this manuscript.  ... 
doi:10.1109/tit.2017.2711601 fatcat:5yab7y44dbexjegtslenjdrjv4

Optimal quaternary linear codes of dimension five

I. Boukliev, R. Daskalov, S. Kapralov
1996 IEEE Transactions on Information Theory  
A survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for drl ( n , 5) is presented. distance of a q-ary  ...  [ rt k , d] -code for given values of n and k 168, dq(232 5) = 172, d4(237,5) 176, d4(240 3 ) = 178, d4(242 3) Index Terms-Minimum distance bounds, quaternary linear codes. c-concatenation sh-shortened  ...  A linear code C of length n and dimension k over GF ( q ) is a k-dimensional subspace of V ( n , 4). Such a code is called [ T I . k . d: q] -code if its minimum Hamming distance is d .  ... 
doi:10.1109/18.508846 fatcat:je2ktxveobhcdneg4tglxmm42i

Page 1380 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
On the minimum distance of ternary cyclic codes; Chapter 3. Five new optimal ternary linear codes; Chapter 4. An optimal ternary [69, 5,45] code and related codes; Chapter 5.  ...  Four non-existence results for ternary linear codes; Chapter 8. Some new results for ternary linear codes of dimension 5 and 6; Chapter 9.  ... 

Short Barcodes for Next Generation Sequencing

Katharina Mir, Klaus Neuhaus, Martin Bossert, Steffen Schober, Eshel Ben-Jacob
2013 PLoS ONE  
We provide a list of codes for lengths between six and eight nucleotides, where for length eight, two substitution errors can be corrected. In fact, no code with larger minimum distance can exist.  ...  We compare published barcodes with codes obtained by two new constructions methods, one based on the currently best known linear codes and a simple randomized construction method.  ...  Acknowledgments The authors would also like to thank R. Landstorfer, V. Sidorenko, C. Senger, and A. Zeh for stimulating discussions.  ... 
doi:10.1371/journal.pone.0082933 pmid:24386128 pmcid:PMC3875435 fatcat:5ui4oyk26zgmxncvpgddfdcmua

Asymmetric Lee Distance Codes for DNA-Based Storage

Ryan Gabrys, Han Mao Kiah, Olgica Milenkovic
2017 IEEE Transactions on Information Theory  
First, we derive upper bounds on the size of the codes under the asymmetric Lee distance measure based on linear programming techniques. Second, we propose code constructions which imply lower bounds.  ...  We consider a new family of asymmetric Lee codes that arise in the design and implementation of DNA-based storage systems and systems with parallel string transmission protocols.  ...  Next, consider the case (b). For a quaternary code C 4 to have minimum ALD distance five, the code should have Hamming distance at least five.  ... 
doi:10.1109/tit.2017.2700847 fatcat:dthk6tdhrjdbrgds2zogmoyf4q
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