Best Known (217−36, 217, s)-Nets in Base 3
(217−36, 217, 1480)-Net over F3 — Constructive and digital
Digital (181, 217, 1480)-net over F3, using
- 3 times m-reduction [i] based on digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 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, 55, 370)-net over F81, using
(217−36, 217, 8418)-Net over F3 — Digital
Digital (181, 217, 8418)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 8418, F3, 2, 36) (dual of [(8418, 2), 16619, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 9846, F3, 2, 36) (dual of [(9846, 2), 19475, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3217, 19692, F3, 36) (dual of [19692, 19475, 37]-code), using
- 1 times truncation [i] based on linear OA(3218, 19693, F3, 37) (dual of [19693, 19475, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3208, 19683, F3, 35) (dual of [19683, 19475, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 1 times truncation [i] based on linear OA(3218, 19693, F3, 37) (dual of [19693, 19475, 38]-code), using
- OOA 2-folding [i] based on linear OA(3217, 19692, F3, 36) (dual of [19692, 19475, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 9846, F3, 2, 36) (dual of [(9846, 2), 19475, 37]-NRT-code), using
(217−36, 217, 2133331)-Net in Base 3 — Upper bound on s
There is no (181, 217, 2133332)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34 301656 099535 840429 177304 943226 912617 800376 818847 471172 773590 996719 545792 335434 916696 673653 794726 541081 > 3217 [i]