Best Known (106, 106+35, s)-Nets in Base 3
(106, 106+35, 328)-Net over F3 — Constructive and digital
Digital (106, 141, 328)-net over F3, using
- 31 times duplication [i] based on digital (105, 140, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 35, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 35, 82)-net over F81, using
(106, 106+35, 667)-Net over F3 — Digital
Digital (106, 141, 667)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 667, F3, 35) (dual of [667, 526, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 749, F3, 35) (dual of [749, 608, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(3136, 729, F3, 35) (dual of [729, 593, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3141, 749, F3, 35) (dual of [749, 608, 36]-code), using
(106, 106+35, 30472)-Net in Base 3 — Upper bound on s
There is no (106, 141, 30473)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 140, 30473)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 266075 835322 804712 683098 495247 120965 146171 363351 658631 646817 499347 > 3140 [i]