Greedy generation of non-binary codes

Monroe, Laura; Pless, Vera

doi:10.1109/isit.1995.535750

Cited by 4 publications

(4 citation statements)

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“…In Coding Theory, so called greedy codes (or lexicographical codes, or lexicodes) are considered, see e.g. [22,59,60,67,82] and references therein. These codes are constructed in two ways.…”

Section: Algorithm Fopmentioning

confidence: 99%

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Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

et al. 2015

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In the projective plane PG(2, q), upper bounds on the smallest size t2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t2(2, q) valid in this region, in particular t2(2, q) < 0.998 3q ln q for 7 ≤ q ≤ 160001;t2(2, q) < 1.05 3q ln q for 7 ≤ q ≤ 301813;t2(2, q) < √ q ln 0.7295 q for 109 ≤ q ≤ 160001;t2(2, q) < √ q ln 0.7404 q for 160001 < q ≤ 301813.The new upper bounds are obtained by finding new small complete arcs in PG(2, q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q's with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs

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Section: Algorithm Fopmentioning

confidence: 99%

“…In another kind, see [22,59,60], one creates a parity check matrix of a q-ary code with codimension r and distance d. All q-ary r-vectors are written as columns in a list in some order. The first column of the list should be included into the matrix.…”

Section: Algorithm Fopmentioning

confidence: 99%

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

et al. 2015

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“…Then, formally, the algorithm FOP can be considered as a version of the recursive g-parity check algorithm for greedy codes of codimension r = 3 and minimum distance d = 4, see e.g. [30, p. 25], [72], [78,Section 7]. However, in coding theory, for given r, d, the aim is to get a long code while our goal is to obtain a short complete arc.…”

Section: Step-by-step Algorithm Fopmentioning

confidence: 99%

Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

Bartoli¹,

Davydov²,

Faina³

et al. 2014

Preprint

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In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane PG(2, q) is proposed to construct small complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that PG(2, q) contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of PG(2, q) is not relevant. A complete lexiarc in PG(2, q) is a complete arc obtained by the algorithm FOP using the lexicographical order of points. In this work, we collect and analyze the sizes of complete lexiarcs in the following regions: all q ≤ 321007, q prime power; 15 sporadic q's in the interval [323761 . . . 430007], see (1.10).

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“…Greedy codes are known to satisfy or be very close to the optimal dimensions for all blocklength-minimum distance pairs [27] and can be generalized to non-binary fields [28] for achieving higher diversity orders with NC as discussed in [17]. Moreover, they are readily available for all dimensions (number of nodes) and minimum distances unlike some other optimal codes.…”

Section: B An Example Of Close-to-optimal Linear Block Codes: Greedymentioning

confidence: 99%

Practical Methods for Wireless Network Coding With Multiple Unicast Transmissions

Aktas

Yılmaz

Aktas

2013

IEEE Trans. Commun.

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We propose a simple yet effective wireless network coding and decoding technique for a multiple unicast network. It utilizes spatial diversity through cooperation between nodes which carry out distributed encoding operations dictated by generator matrices of linear block codes. In order to exemplify the technique, we make use of greedy codes over the binary field and show that the arbitrary diversity orders can be flexibly assigned to nodes. Furthermore, we present the optimal detection rule for the given model that accounts for intermediate node errors and suggest a low-complexity network decoder using the sum-product (SP) algorithm. The proposed SP detector exhibits near optimal performance. We also show asymptotic superiority of network coding over a method that utilizes the wireless channel in a repetitive manner without network coding (NC) and give related rate-diversity trade-off curves. Finally, we extend the given encoding method through selective encoding in order to obtain extra coding gains. Index TermsWireless network coding, cooperative communication, linear block code, sum-product decoding, unequal error protection

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Greedy generation of non-binary codes

Abstract: We get a B-ordering of all binary n-tuples V n by choosing an o:dered basis { y i , ...,yn} o f V n and ordering the n-tuples as follows: 0, y l , y2, y2+y1, y 3 , y 3 + y 1 , y3+y2, y3+y2+yi, y 4 , ... Given a minimum

Cited by 4 publications

References 5 publications

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

Practical Methods for Wireless Network Coding With Multiple Unicast Transmissions

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