Best Known (127−31, 127, s)-Nets in Base 3
(127−31, 127, 328)-Net over F3 — Constructive and digital
Digital (96, 127, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (96, 128, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 82)-net over F81, using
(127−31, 127, 665)-Net over F3 — Digital
Digital (96, 127, 665)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3127, 665, F3, 31) (dual of [665, 538, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 758, F3, 31) (dual of [758, 631, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3126, 757, F3, 31) (dual of [757, 631, 32]-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, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3126, 757, F3, 31) (dual of [757, 631, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 758, F3, 31) (dual of [758, 631, 32]-code), using
(127−31, 127, 32686)-Net in Base 3 — Upper bound on s
There is no (96, 127, 32687)-net in base 3, because
- 1 times m-reduction [i] would yield (96, 126, 32687)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 310584 106396 143447 019589 452872 729115 873433 042645 735382 517411 > 3126 [i]