覆盖半径为 3 且余维数为 4 和 5 的短线性码的表、界与图形
信息论
2020-06-16 v6 组合数学
math.IT
摘要
长度函数 ℓ q ( r , R ) \ell_q(r,R) ℓ q ( r , R ) r r r R R R q q q d d d ℓ q ( r , R , d ) \ell_q(r,R,d) ℓ q ( r , R , d ) r r r R R R d d d q q q q q q R = 3 R=3 R = 3 [ n , n − 4 , 5 ] q 3 [n,n-4,5]_q3 [ n , n − 4 , 5 ] q 3 [ n , n − 5 , 5 ] q 3 [n,n-5,5]_q3 [ n , n − 5 , 5 ] q 3 [ n , n − 5 , 3 ] q 3 [n,n-5,3]_q3 [ n , n − 5 , 3 ] q 3 d d d ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.8 ln q 3 ⋅ q ( 4 − 3 ) / 3 = 2.8 ln q 3 ⋅ q 3 = 2.8 q ln q 3 for 11 ≤ q ≤ 7057 ; \ell_q(4,3)\le\ell_q(4,3,5)<2.8\sqrt[3]{\ln q}\cdot q^{(4-3)/3}=2.8\sqrt[3]{\ln q}\cdot\sqrt[3]{q}=2.8\sqrt[3]{q\ln q}~\text{for}~11\le q\le7057; ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.8 3 ln q ⋅ q ( 4 − 3 ) /3 = 2.8 3 ln q ⋅ 3 q = 2.8 3 q ln q for 11 ≤ q ≤ 7057 ; ℓ q ( 5 , 3 ) ≤ ℓ q ( 5 , 3 , 5 ) < 3 ln q 3 ⋅ q ( 5 − 3 ) / 3 = 3 ln q 3 ⋅ q 2 3 = 3 q 2 ln q 3 for 37 ≤ q ≤ 839. \ell_q(5,3)\le\ell_q(5,3,5)<3\sqrt[3]{\ln q}\cdot q^{(5-3)/3}=3\sqrt[3]{\ln q}\cdot\sqrt[3]{q^2}=3\sqrt[3]{q^2\ln q}~~\text{ for }~37\le q\le839. ℓ q ( 5 , 3 ) ≤ ℓ q ( 5 , 3 , 5 ) < 3 3 ln q ⋅ q ( 5 − 3 ) /3 = 3 3 ln q ⋅ 3 q 2 = 3 3 q 2 ln q for 37 ≤ q ≤ 839. ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.61 q ln q 3 if 13 ≤ q ≤ 4373 ; \ell_q(4,3)\le \ell_q(4,3,5)< 2.61\sqrt[3]{q\ln q}~\text{ if }~13\le q\le4373; ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.61 3 q ln q if 13 ≤ q ≤ 4373 ; ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.65 q ln q 3 if 4373 < q ≤ 7057 ; \ell_q(4,3)\le \ell_q(4,3,5)< 2.65\sqrt[3]{q\ln q}~\text{ if }~4373<q\le7057; ℓ q ( 4 , 3 ) ≤ ℓ q ( 4 , 3 , 5 ) < 2.65 3 q ln q if 4373 < q ≤ 7057 ; ℓ q ( 5 , 3 ) < 2.785 q 2 ln q 3 if 11 ≤ q ≤ 401 ; \ell_q(5,3)<2.785\sqrt[3]{q^2\ln q}~\text{ if }~11\le q\le401; ℓ q ( 5 , 3 ) < 2.785 3 q 2 ln q if 11 ≤ q ≤ 401 ; ℓ q ( 5 , 3 ) ≤ ℓ q ( 5 , 3 , 5 ) < 2.884 q 2 ln q 3 if 401 < q ≤ 839. \ell_q(5,3)\le\ell_q(5,3,5)<2.884\sqrt[3]{q^2\ln q}~\text{ if }~401<q\le839. ℓ q ( 5 , 3 ) ≤ ℓ q ( 5 , 3 , 5 ) < 2.884 3 q 2 ln q if 401 < q ≤ 839. r r r R R R q q q μ q ( r , R ) \mu_q(r,R) μ q ( r , R ) μ q ( 4 , 3 ) < 3.3 ⋅ ln q for 11 ≤ q ≤ 7057 ; \mu_q(4,3)<3.3\cdot\ln q~~\text{ for }~11\le q\le7057; μ q ( 4 , 3 ) < 3.3 ⋅ ln q for 11 ≤ q ≤ 7057 ; μ q ( 5 , 3 ) < 4.2 ⋅ ln q for 37 ≤ q ≤ 839. \mu_q(5,3)<4.2\cdot\ln q~~\text{ for }~37\le q\le839. μ q ( 5 , 3 ) < 4.2 ⋅ ln q for 37 ≤ q ≤ 839.
引用
@article{arxiv.1712.07078,
title = {Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5},
author = {Daniele Bartoli and Alexander A. Davydov and Stefano Marcugini and Fernanda Pambianco},
journal= {arXiv preprint arXiv:1712.07078},
year = {2020}
}
备注
51 pages, 14 figures, 5 tables, 35 references; new results of computer search are added