Best Known (173, 173+33, s)-Nets in Base 3
(173, 173+33, 1480)-Net over F3 — Constructive and digital
Digital (173, 206, 1480)-net over F3, using
- t-expansion [i] based on digital (172, 206, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 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, 52, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (172, 208, 1480)-net over F3, using
(173, 173+33, 9858)-Net over F3 — Digital
Digital (173, 206, 9858)-net over F3, using
- 31 times duplication [i] based on digital (172, 205, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 9858, F3, 2, 33) (dual of [(9858, 2), 19511, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3205, 19716, F3, 33) (dual of [19716, 19511, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(3205, 19716, F3, 33) (dual of [19716, 19511, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 9858, F3, 2, 33) (dual of [(9858, 2), 19511, 34]-NRT-code), using
(173, 173+33, 4411875)-Net in Base 3 — Upper bound on s
There is no (173, 206, 4411876)-net in base 3, because
- 1 times m-reduction [i] would yield (173, 205, 4411876)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 64 544243 079465 945264 505603 814915 157119 966858 754741 717436 334047 057304 255666 715220 895766 654708 974721 > 3205 [i]