Best Known (140, 140+36, s)-Nets in Base 3
(140, 140+36, 688)-Net over F3 — Constructive and digital
Digital (140, 176, 688)-net over F3, using
- t-expansion [i] based on digital (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
(140, 140+36, 1901)-Net over F3 — Digital
Digital (140, 176, 1901)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 1901, F3, 36) (dual of [1901, 1725, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 2216, F3, 36) (dual of [2216, 2040, 37]-code), using
- construction XX applied to Ce(36) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3176, 2216, F3, 36) (dual of [2216, 2040, 37]-code), using
(140, 140+36, 174679)-Net in Base 3 — Upper bound on s
There is no (140, 176, 174680)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 940501 061919 300217 420544 539522 302039 691281 338914 975439 579044 823433 880101 938606 627217 > 3176 [i]