Best Known (113−20, 113, s)-Nets in Base 3
(113−20, 113, 688)-Net over F3 — Constructive and digital
Digital (93, 113, 688)-net over F3, using
- 31 times duplication [i] based on digital (92, 112, 688)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (91, 112, 688)-net over F3, using
(113−20, 113, 3498)-Net over F3 — Digital
Digital (93, 113, 3498)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3113, 3498, F3, 20) (dual of [3498, 3385, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 6594, F3, 20) (dual of [6594, 6481, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(13) [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(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3113, 6594, F3, 20) (dual of [6594, 6481, 21]-code), using
(113−20, 113, 557710)-Net in Base 3 — Upper bound on s
There is no (93, 113, 557711)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 821680 543878 439360 521086 801490 663868 454548 986675 952397 > 3113 [i]