Best Known (136, 136+18, s)-Nets in Base 3
(136, 136+18, 59056)-Net over F3 — Constructive and digital
Digital (136, 154, 59056)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (126, 144, 59049)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 59049, F3, 18, 18) (dual of [(59049, 18), 1062738, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3144, 531441, F3, 18) (dual of [531441, 531297, 19]-code), using
- 1 times truncation [i] based on linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(3144, 531441, F3, 18) (dual of [531441, 531297, 19]-code), using
- net defined by OOA [i] based on linear OOA(3144, 59049, F3, 18, 18) (dual of [(59049, 18), 1062738, 19]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(136, 136+18, 219552)-Net over F3 — Digital
Digital (136, 154, 219552)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3154, 219552, F3, 2, 18) (dual of [(219552, 2), 438950, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3154, 265749, F3, 2, 18) (dual of [(265749, 2), 531344, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3154, 531498, F3, 18) (dual of [531498, 531344, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 531499, F3, 18) (dual of [531499, 531345, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(397, 531442, F3, 13) (dual of [531442, 531345, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3154, 531499, F3, 18) (dual of [531499, 531345, 19]-code), using
- OOA 2-folding [i] based on linear OA(3154, 531498, F3, 18) (dual of [531498, 531344, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3154, 265749, F3, 2, 18) (dual of [(265749, 2), 531344, 19]-NRT-code), using
(136, 136+18, large)-Net in Base 3 — Upper bound on s
There is no (136, 154, large)-net in base 3, because
- 16 times m-reduction [i] would yield (136, 138, large)-net in base 3, but