Best Known (178, 178+29, s)-Nets in Base 3
(178, 178+29, 4226)-Net over F3 — Constructive and digital
Digital (178, 207, 4226)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 16, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-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 (2, 16, 8)-net over F3, using
(178, 178+29, 29557)-Net over F3 — Digital
Digital (178, 207, 29557)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3207, 29557, F3, 2, 29) (dual of [(29557, 2), 58907, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3206, 29557, F3, 2, 29) (dual of [(29557, 2), 58908, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3206, 59114, F3, 29) (dual of [59114, 58908, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [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(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(315, 65, F3, 6) (dual of [65, 50, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3206, 59114, F3, 29) (dual of [59114, 58908, 30]-code), using
- 31 times duplication [i] based on linear OOA(3206, 29557, F3, 2, 29) (dual of [(29557, 2), 58908, 30]-NRT-code), using
(178, 178+29, large)-Net in Base 3 — Upper bound on s
There is no (178, 207, large)-net in base 3, because
- 27 times m-reduction [i] would yield (178, 180, large)-net in base 3, but