Generalized Hamming weights of projective Reed–Muller-type codes over graphs

Martinez-Bernal, Jose; Valencia-Bucio, Miguel A.; Villarreal, Rafael H.

doi:10.1016/j.disc.2019.111639

Cited by 4 publications

(3 citation statements)

References 30 publications

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“…Secondly, we introduce rotated LBP descriptors [12] to enrich the feature point neighbor information. Finally, the similarity is calculated by using Hamming distance [13] to select the four points to be matched with high similarity, and then the cosine similarity [14] is used to screen out the points to be matched with large disparity in directional features.…”

Section: Eai Endorsed Transactions On Internet Of Thingsmentioning

confidence: 99%

Small-area Fingerprint Recognition Based on Improved ORB Algorithm in Embedded Environment

Xiao

Liu

2022

EAI Endorsed Trans IoT

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Most of the fingerprint matching algorithms were proposed for large area fingerprints, which can hardly work effectively in small-area fingerprints. In this work, an improved ORB algorithm is proposed for small-area fingerprint matching in embedded mobile devices. In feature descriptor design, we analyzed the characters of the fingerprint in the embedded mobile devices and discard the multi-scale feature process to reduce the amount of operations. Moreover, we proposed a fusion descriptor combing LBP and rBRIEF descriptor. In the key point matching process, we proposed a two-step (coarse and fine) matching method by using Hamming distance and cosine similarity, respectively. The experimental results show that the proposed method has a rejection rate of 6.4%, a false recognition rate of 0.1%, and an average matching time of 58ms. It can effectively improve the performance of small-area fingerprint matching and meet the application requirements of embedded mobile device authentication.

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Section: Eai Endorsed Transactions On Internet Of Thingsmentioning

confidence: 99%

Small-area Fingerprint Recognition Based on Improved ORB Algorithm in Embedded Environment

Xiao

Liu

2022

EAI Endorsed Trans IoT

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“…If r = 1, υ 1 (G − ) is denoted υ(G − ). For simple graphs, the following combinatorial formulas for the generalized Hamming weights were shown in [26]. Proof.…”

Section: Generalized Hamming Weights Over Signed Graphsmentioning

confidence: 99%

“…We show that the formulas of [26,Corollary 2.13] for the generalized Hamming weights of incidence matrix codes of simple graphs can be extended to multigraphs (Corollary 3.17). Then we show combinatorial formulas for the minimum distance of the incidence matrix code of a signed graph [28, Proposition 9.2.4] (Corollary 3.18).…”

Section: Introductionmentioning

confidence: 99%

Linear codes over signed graphs

Martinez-Bernal

Valencia-Bucio

Villarreal

2019

Des. Codes Cryptogr.

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We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual code. Then we determine the regularity of the ideals of circuits and cocircuits of a signed graph, and prove an algebraic formula in terms of the multiplicity for the frustration index of an unbalanced signed graph. IntroductionThe generalized Hamming weights (GHWs) of a linear code are parameters of interest in many applications [12,16,20,27,31,37,42,43,45] and they have been nicely related to the graded Betti numbers of the ideal of cocircuits of the matroid of a linear code [19,20], to the nullity function of the dual matroid of a linear code [42], and to the enumerative combinatorics of linear codes [3,18,22,23]. Because of this, their study has attracted considerable attention, but determining them is in general a difficult problem. The notion of generalized Hamming weight was introduced by Helleseth, Kløve and Mykkeltveit in [17] and was first used systematically by Wei in [42]. For convenience we recall this notion. Let K = F q be a finite field and let C be an [m, k]-linear code of length m and dimension k, that is, C is a linear subspace of K m with k = dim K (C). Let 1 ≤ r ≤ k be an integer. Given a linear subspace D of C, the support of D, denoted χ(D), is the set of nonzero positions of D, that is, χ(D) := {i : ∃ (a 1 , . . . , a m ) ∈ D, a i = 0}.The r-th generalized Hamming weight of C, denoted δ r (C), is given by δ r (C) := min{|χ(D)| : D is a subspace of C, dim K (D) = r}.As usual we call the set {δ 1 (C), . . . , δ k (C)} the weight hierarchy of the linear code C. The 1st Hamming weight of C is the minimum distance δ(C) of C, that is, one haswhere ω(x) is the Hamming weight of the vector x, i.e., the number of non-zero entries of x. To determine the minimum distance is essential to find good error-correcting codes [23].The notion of generalized Hamming weights for linear codes was extended to matroids by Britz, Johnsen, Mayhew and Shiromoto [6, p. 332] as we now explain.Let M be a matroid with ground set E, rank function ρ, nullity function η, and let M * be its dual matroid. The r-th generalized Hamming weight of M , denoted d r (M ), is given by d r (M ) := min{|X| : X ⊆ E and η(X) = r} for 1 ≤ r ≤ η(E).2010 Mathematics Subject Classification. Primary 94B05; Secondary 94C15, 05C40, 05C22; 13P25.

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