Best Known (125−26, 125, s)-Nets in Base 3
(125−26, 125, 640)-Net over F3 — Constructive and digital
Digital (99, 125, 640)-net over F3, using
- 31 times duplication [i] based on digital (98, 124, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 31, 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, 31, 160)-net over F81, using
(125−26, 125, 1408)-Net over F3 — Digital
Digital (99, 125, 1408)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 1408, F3, 26) (dual of [1408, 1283, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 2207, F3, 26) (dual of [2207, 2082, 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(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3125, 2207, F3, 26) (dual of [2207, 2082, 27]-code), using
(125−26, 125, 109655)-Net in Base 3 — Upper bound on s
There is no (99, 125, 109656)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 436691 099993 430215 788181 682305 451455 170768 471982 916216 403377 > 3125 [i]