Best Known (140, 178, s)-Nets in Base 3
(140, 178, 640)-Net over F3 — Constructive and digital
Digital (140, 178, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (140, 180, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 45, 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, 45, 160)-net over F81, using
(140, 178, 1550)-Net over F3 — Digital
Digital (140, 178, 1550)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3178, 1550, F3, 38) (dual of [1550, 1372, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 2197, F3, 38) (dual of [2197, 2019, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(37) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3178, 2197, F3, 38) (dual of [2197, 2019, 39]-code), using
(140, 178, 116947)-Net in Base 3 — Upper bound on s
There is no (140, 178, 116948)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8 465324 464009 867383 339552 873799 905804 469617 135878 660793 963850 032763 569297 295199 138289 > 3178 [i]