Best Known (176, 176+29, s)-Nets in Base 3
(176, 176+29, 4222)-Net over F3 — Constructive and digital
Digital (176, 205, 4222)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (162, 191, 4218)-net over F3, using
- net defined by OOA [i] based on linear OOA(3191, 4218, F3, 29, 29) (dual of [(4218, 29), 122131, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3191, 59053, F3, 29) (dual of [59053, 58862, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3191, 59053, F3, 29) (dual of [59053, 58862, 30]-code), using
- net defined by OOA [i] based on linear OOA(3191, 4218, F3, 29, 29) (dual of [(4218, 29), 122131, 30]-NRT-code), using
- digital (0, 14, 4)-net over F3, using
(176, 176+29, 27996)-Net over F3 — Digital
Digital (176, 205, 27996)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 27996, F3, 2, 29) (dual of [(27996, 2), 55787, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 29551, F3, 2, 29) (dual of [(29551, 2), 58897, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3204, 29551, F3, 2, 29) (dual of [(29551, 2), 58898, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3202, 29550, F3, 2, 29) (dual of [(29550, 2), 58898, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3202, 59100, F3, 29) (dual of [59100, 58898, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3202, 59100, F3, 29) (dual of [59100, 58898, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3202, 29550, F3, 2, 29) (dual of [(29550, 2), 58898, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3204, 29551, F3, 2, 29) (dual of [(29551, 2), 58898, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 29551, F3, 2, 29) (dual of [(29551, 2), 58897, 30]-NRT-code), using
(176, 176+29, large)-Net in Base 3 — Upper bound on s
There is no (176, 205, large)-net in base 3, because
- 27 times m-reduction [i] would yield (176, 178, large)-net in base 3, but