Optimal minimal linear codes from posets
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Optimal minimal linear codes from posets Jong Yoon Hyun1 · Hyun Kwang Kim2 · Yansheng Wu3 · Qin Yue4,5 Received: 19 April 2019 / Revised: 5 March 2020 / Accepted: 17 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun et al. We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the condition of Ashikhmin–Barg. Keywords Poset · Optimal binary linear code · Minimal binary linear code · Weight distribution Mathematics Subject Classification 94B05 · 06A11 · 06E30
Communicated by J.-L. Kim.
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Qin Yue [email protected] Jong Yoon Hyun [email protected] Hyun Kwang Kim [email protected] Yansheng Wu [email protected]
1
Konkuk University, Glocal Campus 268 Chungwon-daero, Chungju-si, Chungcheongbuk-do 27478, South Korea
2
Pohang University of Science and Technology, 77 Cheongam-ro, Nam-Gu, Pohang 37673, South Korea
3
School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. China
4
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, Jiangsu, P. R. China
5
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, P. R. China
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J. Y. Hyun et al.
1 Introduction Let F2 be the finite field with order two. For positive integers n, k and d, an [n, k, d] binary linear code C is a k-dimensional subspace of Fn2 with minimum (Hamming) distance d. We sometimes denote by wmin instead of d. The support supp(v) of a vector v ∈ Fn2 is defined by the set of nonzero coordinate positions. The Hamming weight wt(v) of v ∈ Fn2 is defined by the size of supp(v). We say that a linear code is distance-optimal if it has the highest minimum distance with prescribed length and dimension. An [n, k, d] linear code is called almost distance-optimal if the code [n, k, d + 1] is optimal, see [14, Chapter 2]. For an [n, k, d] binary linear code, the Griesmer bound (see [11]) states that n≥
k−1 d i=0
2i
,
where x denotes the smallest integer greater than or equal to x. We say that a linear code is a Griesmer code if it meets the Griesmer bound with equality. One can verify that Griesmer codes are distance-optimal. Let Ai be the number of codewords in a linear code C with Hamming weight i. The weight enumerator of C is defined by 1+ A1 z+ A2 z 2 +· · ·+ An z n . The sequence (1, A1 , A2 , . . . , An ) is called the weight distribution of C . A code C is t-weight if the number of nonzero Ai in the sequence (A1 , A2 , . . . , An ) is equal to t. The study of the weight distributio
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