Best Known (137, 137+19, s)-Nets in Base 3
(137, 137+19, 59057)-Net over F3 — Constructive and digital
Digital (137, 156, 59057)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 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 (126, 145, 59049)-net over F3, using
- net defined by OOA [i] based on linear OOA(3145, 59049, F3, 19, 19) (dual of [(59049, 19), 1121786, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using
- net defined by OOA [i] based on linear OOA(3145, 59049, F3, 19, 19) (dual of [(59049, 19), 1121786, 20]-NRT-code), using
- digital (2, 11, 8)-net over F3, using
(137, 137+19, 177167)-Net over F3 — Digital
Digital (137, 156, 177167)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3156, 177167, F3, 3, 19) (dual of [(177167, 3), 531345, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3156, 531501, F3, 19) (dual of [531501, 531345, 20]-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(311, 59, F3, 5) (dual of [59, 48, 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 C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 3-folding [i] based on linear OA(3156, 531501, F3, 19) (dual of [531501, 531345, 20]-code), using
(137, 137+19, large)-Net in Base 3 — Upper bound on s
There is no (137, 156, large)-net in base 3, because
- 17 times m-reduction [i] would yield (137, 139, large)-net in base 3, but