Best Known (126−30, 126, s)-Nets in Base 3
(126−30, 126, 400)-Net over F3 — Constructive and digital
Digital (96, 126, 400)-net over F3, using
- 32 times duplication [i] based on digital (94, 124, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 31, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 31, 100)-net over F81, using
(126−30, 126, 737)-Net over F3 — Digital
Digital (96, 126, 737)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3126, 737, F3, 30) (dual of [737, 611, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3126, 758, F3, 30) (dual of [758, 632, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 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(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(38, 29, F3, 4) (dual of [29, 21, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3126, 758, F3, 30) (dual of [758, 632, 31]-code), using
(126−30, 126, 32686)-Net in Base 3 — Upper bound on s
There is no (96, 126, 32687)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 310584 106396 143447 019589 452872 729115 873433 042645 735382 517411 > 3126 [i]