Best Known (127−26, 127, s)-Nets in Base 3
(127−26, 127, 640)-Net over F3 — Constructive and digital
Digital (101, 127, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (101, 128, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 32, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 32, 160)-net over F81, using
(127−26, 127, 1545)-Net over F3 — Digital
Digital (101, 127, 1545)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3127, 1545, F3, 26) (dual of [1545, 1418, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 2215, F3, 26) (dual of [2215, 2088, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3126, 2214, F3, 26) (dual of [2214, 2088, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-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
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3126, 2214, F3, 26) (dual of [2214, 2088, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 2215, F3, 26) (dual of [2215, 2088, 27]-code), using
(127−26, 127, 129850)-Net in Base 3 — Upper bound on s
There is no (101, 127, 129851)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 930447 061212 294060 772890 433674 416452 274716 208775 658657 001823 > 3127 [i]