Best Known (186, 186+23, s)-Nets in Base 3
(186, 186+23, 144947)-Net over F3 — Constructive and digital
Digital (186, 209, 144947)-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 (173, 196, 144939)-net over F3, using
- net defined by OOA [i] based on linear OOA(3196, 144939, F3, 23, 23) (dual of [(144939, 23), 3333401, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3196, 1594330, F3, 23) (dual of [1594330, 1594134, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3196, 1594330, F3, 23) (dual of [1594330, 1594134, 24]-code), using
- net defined by OOA [i] based on linear OOA(3196, 144939, F3, 23, 23) (dual of [(144939, 23), 3333401, 24]-NRT-code), using
- digital (2, 13, 8)-net over F3, using
(186, 186+23, 531462)-Net over F3 — Digital
Digital (186, 209, 531462)-net over F3, using
- 31 times duplication [i] based on digital (185, 208, 531462)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 531462, F3, 3, 23) (dual of [(531462, 3), 1594178, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3207, 531462, F3, 3, 23) (dual of [(531462, 3), 1594179, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3207, 1594386, F3, 23) (dual of [1594386, 1594179, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(311, 63, F3, 5) (dual of [63, 52, 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(22) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3207, 1594386, F3, 23) (dual of [1594386, 1594179, 24]-code), using
- 31 times duplication [i] based on linear OOA(3207, 531462, F3, 3, 23) (dual of [(531462, 3), 1594179, 24]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 531462, F3, 3, 23) (dual of [(531462, 3), 1594178, 24]-NRT-code), using
(186, 186+23, large)-Net in Base 3 — Upper bound on s
There is no (186, 209, large)-net in base 3, because
- 21 times m-reduction [i] would yield (186, 188, large)-net in base 3, but