Best Known (109−20, 109, s)-Nets in Base 3
(109−20, 109, 688)-Net over F3 — Constructive and digital
Digital (89, 109, 688)-net over F3, using
- 31 times duplication [i] based on digital (88, 108, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 27, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 27, 172)-net over F81, using
(109−20, 109, 3290)-Net over F3 — Digital
Digital (89, 109, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 3290, F3, 2, 20) (dual of [(3290, 2), 6471, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3109, 6580, F3, 20) (dual of [6580, 6471, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3109, 6581, F3, 20) (dual of [6581, 6472, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3109, 6581, F3, 20) (dual of [6581, 6472, 21]-code), using
- OOA 2-folding [i] based on linear OA(3109, 6580, F3, 20) (dual of [6580, 6471, 21]-code), using
(109−20, 109, 359382)-Net in Base 3 — Upper bound on s
There is no (89, 109, 359383)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10144 415295 783673 363510 379415 854846 329455 671735 802141 > 3109 [i]