Best Known (91, 91+25, s)-Nets in Base 3
(91, 91+25, 600)-Net over F3 — Constructive and digital
Digital (91, 116, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 29, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(91, 91+25, 1125)-Net over F3 — Digital
Digital (91, 116, 1125)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3116, 1125, F3, 25) (dual of [1125, 1009, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3116, 2192, F3, 25) (dual of [2192, 2076, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(399, 2188, F3, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3116, 2192, F3, 25) (dual of [2192, 2076, 26]-code), using
(91, 91+25, 98785)-Net in Base 3 — Upper bound on s
There is no (91, 116, 98786)-net in base 3, because
- 1 times m-reduction [i] would yield (91, 115, 98786)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7 395617 081594 677552 682380 623223 098606 651363 960599 489913 > 3115 [i]