关于半径为 R、余维数为 tR+1 的覆盖码的进一步结果
组合数学
2024-03-04 v2
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 [ n , n − r ] q R [n,n-r]_qR [ n , n − r ] q R n n n s q ( N , ρ ) s_q(N,\rho) s q ( N , ρ ) P G ( N , q ) \mathrm{PG}(N,q) PG ( N , q ) ρ \rho ρ [ n , n − r ] q R [n,n-r]_qR [ n , n − r ] q R P G ( r − 1 , q ) \mathrm{PG}(r-1,q) PG ( r − 1 , q ) ( R − 1 ) (R-1) ( R − 1 ) n n n ℓ q ( r , R ) = s q ( r − 1 , R − 1 ) \ell_q(r,R)=s_q(r-1,R-1) ℓ q ( r , R ) = s q ( r − 1 , R − 1 ) R ≥ 3 R\ge3 R ≥ 3 ℓ q ( t R + 1 , R ) \ell_q(tR+1,R) ℓ q ( tR + 1 , R ) ∙ ℓ q ( t R + 1 , R ) = s q ( t R , R − 1 ) ≤ R ! R R − 2 R ⋅ q ( r − R ) / R ⋅ ln q R + o ( q ( r − R ) / R ) , r = t R + 1 , t ≥ 1 , q 为任意素数幂 , q 足够大 ; \hspace{0.7cm} \bullet~\ell_q(tR+1,R) =s_q(tR,R-1)\le \sqrt[R]{\frac{R!}{R^{R-2}}}\cdot q^{(r-R)/R}\cdot\sqrt[R]{\ln q}+o(q^{(r-R)/R}), \hspace{0.3cm}r=tR+1,~t\ge1,~ q\text{ 为任意素数幂},~q\text{ 足够大}; ∙ ℓ q ( tR + 1 , R ) = s q ( tR , R − 1 ) ≤ R R R − 2 R ! ⋅ q ( r − R ) / R ⋅ R ln q + o ( q ( r − R ) / R ) , r = tR + 1 , t ≥ 1 , q 为任意素数幂 , q 足够大 ; ∙ 若此外 R 足够大, 则 R ! R R − 2 R ∼ 1 e ≈ 0.3679. \hspace{0.7cm} \bullet~\text{ 若此外 }R\text{ 足够大, 则 }\sqrt[R]{\frac{R!}{R^{R-2}}}\thicksim\frac{1}{e}\thickapprox0.3679. ∙ 若此外 R 足够大 , 则 R R R − 2 R ! ∼ e 1 ≈ 0.3679. t = 1 t=1 t = 1 P G ( R , q ) \mathrm{PG}(R,q) PG ( R , q ) ( R − 1 ) (R-1) ( R − 1 ) [ n , n − ( R + 1 ) ] q R [n,n-(R+1)]_qR [ n , n − ( R + 1 ) ] q R R + 1 R + 1 R + 1 q m q^m q m r = t R + 1 r=tR+1 r = tR + 1 t ≥ 1 t\ge1 t ≥ 1
引用
@article{arxiv.2310.02715,
title = {Further results on covering codes with radius R and codimension tR + 1},
author = {Alexander A. Davydov and Stefano Marcugini and Fernanda Pambianco},
journal= {arXiv preprint arXiv:2310.02715},
year = {2024}
}
备注
24 pages. arXiv admin note: text overlap with arXiv:2108.13609