Best Known (132−19, 132, s)-Nets in Base 3
(132−19, 132, 6569)-Net over F3 — Constructive and digital
Digital (113, 132, 6569)-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 (102, 121, 6561)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 6561, F3, 19, 19) (dual of [(6561, 19), 124538, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, 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(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- net defined by OOA [i] based on linear OOA(3121, 6561, F3, 19, 19) (dual of [(6561, 19), 124538, 20]-NRT-code), using
- digital (2, 11, 8)-net over F3, using
(132−19, 132, 25578)-Net over F3 — Digital
Digital (113, 132, 25578)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3132, 25578, F3, 2, 19) (dual of [(25578, 2), 51024, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 29550, F3, 2, 19) (dual of [(29550, 2), 58968, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3132, 59100, F3, 19) (dual of [59100, 58968, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3132, 59101, F3, 19) (dual of [59101, 58969, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 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
- discarding factors / shortening the dual code based on linear OA(3132, 59101, F3, 19) (dual of [59101, 58969, 20]-code), using
- OOA 2-folding [i] based on linear OA(3132, 59100, F3, 19) (dual of [59100, 58968, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 29550, F3, 2, 19) (dual of [(29550, 2), 58968, 20]-NRT-code), using
(132−19, 132, large)-Net in Base 3 — Upper bound on s
There is no (113, 132, large)-net in base 3, because
- 17 times m-reduction [i] would yield (113, 115, large)-net in base 3, but