Best Known (92, 92+21, s)-Nets in Base 3
(92, 92+21, 688)-Net over F3 — Constructive and digital
Digital (92, 113, 688)-net over F3, using
- 31 times duplication [i] based on digital (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 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, 28, 172)-net over F81, using
(92, 92+21, 3284)-Net over F3 — Digital
Digital (92, 113, 3284)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3113, 3284, F3, 2, 21) (dual of [(3284, 2), 6455, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3113, 6568, F3, 21) (dual of [6568, 6455, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 6569, F3, 21) (dual of [6569, 6456, 22]-code), using
- 1 times truncation [i] based on linear OA(3114, 6570, F3, 22) (dual of [6570, 6456, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 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(21) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(3114, 6570, F3, 22) (dual of [6570, 6456, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 6569, F3, 21) (dual of [6569, 6456, 22]-code), using
- OOA 2-folding [i] based on linear OA(3113, 6568, F3, 21) (dual of [6568, 6455, 22]-code), using
(92, 92+21, 499684)-Net in Base 3 — Upper bound on s
There is no (92, 113, 499685)-net in base 3, because
- 1 times m-reduction [i] would yield (92, 112, 499685)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 273894 061561 605509 733684 654215 432936 138377 385354 200241 > 3112 [i]