Best Known (249−44, 249, s)-Nets in Base 3
(249−44, 249, 1480)-Net over F3 — Constructive and digital
Digital (205, 249, 1480)-net over F3, using
- 31 times duplication [i] based on digital (204, 248, 1480)-net over F3, using
- t-expansion [i] based on digital (202, 248, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 62, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 62, 370)-net over F81, using
- t-expansion [i] based on digital (202, 248, 1480)-net over F3, using
(249−44, 249, 5381)-Net over F3 — Digital
Digital (205, 249, 5381)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 5381, F3, 44) (dual of [5381, 5132, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 6602, F3, 44) (dual of [6602, 6353, 45]-code), using
- construction X applied to C([0,22]) ⊂ C([0,19]) [i] based on
- linear OA(3241, 6562, F3, 45) (dual of [6562, 6321, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,22]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 6602, F3, 44) (dual of [6602, 6353, 45]-code), using
(249−44, 249, 1137522)-Net in Base 3 — Upper bound on s
There is no (205, 249, 1137523)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 63562 065319 756786 935323 690037 643978 455726 029085 210824 020788 108205 504910 644695 086806 640252 230488 238799 901317 829182 874717 > 3249 [i]