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Tuesday, August 4, 2020 | History

2 edition of Optimal linear codes over GF(3) and GF(4). found in the catalog.

Optimal linear codes over GF(3) and GF(4).

Philip Peter Greenough

# Optimal linear codes over GF(3) and GF(4).

## by Philip Peter Greenough

Written in English

Edition Notes

PhD thesis, Mathematics and Computer Science.

 ID Numbers Series DX182645 Open Library OL21515557M

A linear code with parameters of the form \$[n, k, n-k+1]\$ is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form \$[n, k, n-k]\$ is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost . The desired blocklength of a linear block code is often diﬀerent from the We’ll see that no 4columns are linearly dependent over GF(2), so d∗ ≥ 5. EE , Octo No Page 7. which is essentially optimal. EE , Octo No Page 8. Expanded codes.

We show that codes over GF(16) are ≈ 4 times as complex to decode in return for a dB gain in performance and codes over GF() are ≈ 44 times as complex to decode in return for a ≈ dB gain in performance, all relative to codes over GF(2) at the same blocklength. This provides a quantitative trade-off in selecting. Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q] is k dimensional then we talk about the subspace as an [n,k]-code. (Note that the square Over the field F = GF.

optimal codes can be found that are linear. In this paper we treat the case of M= 23 = 8 and ﬁnd that the optimal codes cannot be linear! Note that for an exact performance analysis, dimensional subspace of the n-dimensional vector space over the channel input alphabet. One of the important properties of a linear code concerns their. David P. Woodruff. Associate Professor of. Computer Science. Research interests: Data streams, machine learning, numerical linear algebra, sketching, and sparse recovery.. Contact: dwoodruf (at) cs (dot) cmu (dot) edu CV (updated May, ).

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### Optimal linear codes over GF(3) and GF(4) by Philip Peter Greenough Download PDF EPUB FB2

The class of quasi-cyclic (QC) codes has been proven to contain many good codes. In this paper, new rate 1/p QC codes over GF(5) are constructed using integer linear programming and heuristic. Discrete Mathematics () North-Holland Optimal linear codes over GF (4) P.P.

Greenough* and R. Hill Department of Mathematics and Computer Science, University of Salford, Salford MS 4WT, UK Received 12 July Revised 14 November Abstract A central problem in coding theory is that of finding the smallest length for which there exists a linear code of Cited by:   Landjev I.N.

() Optimal linear codes of dimension 4 over GF(5). In: Mora T., Mattson H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. In: Mora T., Mattson H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting by: 5.

Example CodeFld_BestLength-GF2 (HE40) We search for best known codes using dimension and minimum weight, looking at codes over GF(2) of dimension Even though the database over GF(2) is % filled up to lengththe code required may be longer than that so we have to check the second boolean return value.

Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets F 3, F   A code which Optimal linear codes over GF book one of these two values is called optimal.

For the case of linear codes over GF(5), Problem 2 has been solved for k⩽3 (see). In addition, n 5 (4,d) has been solved for all but 22 values of d in. After,, there are eight unsolved cases in this dimension. A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a filed of q elements.

We consider here the problem for quaternary codes (q = 4), solving the. Optimal linear codes over GF(4) P.P. Greenough* and R. Hill Department of Mathematics and Computer Science, University of Salford, Salford MS 4WT, UK Received 12 July Revised 14 November Abstract A central problem in coding theory is.

Optimal linear codes over \$GF\$(5)(Semigroups, Formal. Code Tables Alternative address: Linear Block Codes Bounds on the minimum distance of linear block codes over GF(q) with q≤9 for given length and dimension including information on the construction.

query form; Quantum Error-Correcting Codes Bounds on the minimum distance of qubit block codes for given length and. Some high-rate linear codes over GF(5) and GF(7) R. Daskalov 1 Problems of Information Transmission vol pages – () Cite this article. distance optimal indecomposable linear codes over GF(16) Anton Betten.

k=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16; n=1: 2: 3: 4: 4 1: 5: 5 1: 4 3: 6: 5 4: 4 7: 5. The idea of some properties of linear codes over GF (4) due to Carry Huffman and Vera Pless is considered in the context of dual codes over GF(4).In  the minimum weight d of a code is specified,we call it an [n, k, d] Code n is the length of the code k is its dimension and d the.

codes. A linear code C is called optimal in this context if there is no linear code of the same length and higher minimum distance.

There are bounds  giving limits for the optimal minimum distance of a linear code of ﬁxed length n. There is a lower bound, saying there is a known linear code having this minimum distance.

Abstract. The paper discusses weight distribution of periodic errors and then the optimal case on bounds of parity check digits for (,) linear codes over that corrects all periodic errors of order in the first block of length and all periodic errors of order in the second block of length and no r, we extend the study to the case when the errors are in the form of periodic errors.

Abstract: In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes.

These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract.

Let [n, k, d]q-code be a linear code of length n, dimension k and min-imum Hamming distance d over GF (q). One of the most important problems in coding theory is to construct codes with best possible minimum distances. Re-cently, the class of quasi-cyclic (QC) codes has been proven to contain many such codes.

The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol . They introduced a graph parameter, minrk 2(G), which completely characterizes the length of an optimal linear index code for G. The authors of  showed that in various cases linear codes attain the optimal word length, and conjectured that linear index.

As was the case with the (r,d) codes introduced in , a class of pyramid codes turns out to provide examples of optimal (r,δ)i codes, i.e., (r,δ)i codes is which the bound in (2) is achieved with equality. For the special case when r|k, we will identify conditions that the parity check matrix of an optimal (r,δ)i code must necessarily satisfy.

Definition and parameters. A linear code of length n and rank k is a linear subspace C with dimension k of the vector space where is the finite field with q elements. Such a code is called a q-ary q = 2 or q = 3, the code is described as a binary code, or a ternary code respectively.

The vectors in C are called size of a code is the number of codewords and equals q k. In this correspondence we study optimal linear codes of dimension 4 over Fs. We solve the problem of finding n5(4, d) for all but 22 values of d.

In Sections I1 and I11 we give lower bounds on (4, d) which improve the Griesmer bound, i.e., the nonexistence of some four-dimensional codes over .Optimal linear codes over GF(3) and GF(4).

Author: Greenough, Philip Peter. ISNI: Awarding Body: University of Salford Current Institution: University of Salford Date of Award: Availability of Full Text.kyoto university optimal linear code row vector co ordinate place non-zero entry row vector space weight wt galois field ordered mathrm tuples Powered by: About CiteSeerX.