Best Known (188, 188+22, s)-Nets in Base 3
(188, 188+22, 434823)-Net over F3 — Constructive and digital
Digital (188, 210, 434823)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 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 (175, 197, 434815)-net over F3, using
- net defined by OOA [i] based on linear OOA(3197, 434815, F3, 22, 22) (dual of [(434815, 22), 9565733, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3197, 4782965, F3, 22) (dual of [4782965, 4782768, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3197, 4782965, F3, 22) (dual of [4782965, 4782768, 23]-code), using
- net defined by OOA [i] based on linear OOA(3197, 434815, F3, 22, 22) (dual of [(434815, 22), 9565733, 23]-NRT-code), using
- digital (2, 13, 8)-net over F3, using
(188, 188+22, 1195759)-Net over F3 — Digital
Digital (188, 210, 1195759)-net over F3, using
- 32 times duplication [i] based on digital (186, 208, 1195759)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 1195759, F3, 4, 22) (dual of [(1195759, 4), 4782828, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3208, 4783036, F3, 22) (dual of [4783036, 4782828, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(311, 67, F3, 5) (dual of [67, 56, 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(21) ⊂ Ce(15) [i] based on
- OOA 4-folding [i] based on linear OA(3208, 4783036, F3, 22) (dual of [4783036, 4782828, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 1195759, F3, 4, 22) (dual of [(1195759, 4), 4782828, 23]-NRT-code), using
(188, 188+22, large)-Net in Base 3 — Upper bound on s
There is no (188, 210, large)-net in base 3, because
- 20 times m-reduction [i] would yield (188, 190, large)-net in base 3, but