Best Known (141, 178, s)-Nets in Base 3
(141, 178, 688)-Net over F3 — Constructive and digital
Digital (141, 178, 688)-net over F3, using
- 32 times duplication [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
(141, 178, 1767)-Net over F3 — Digital
Digital (141, 178, 1767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3178, 1767, F3, 37) (dual of [1767, 1589, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 2218, F3, 37) (dual of [2218, 2040, 38]-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(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), 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(3178, 2218, F3, 37) (dual of [2218, 2040, 38]-code), using
(141, 178, 185674)-Net in Base 3 — Upper bound on s
There is no (141, 178, 185675)-net in base 3, because
- 1 times m-reduction [i] would yield (141, 177, 185675)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 821612 759693 769910 743097 093248 650510 049286 679033 932276 792700 429060 741710 391440 712981 > 3177 [i]