Best Known (107−19, 107, s)-Nets in Base 3
(107−19, 107, 736)-Net over F3 — Constructive and digital
Digital (88, 107, 736)-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 (78, 97, 729)-net over F3, using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, 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(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using
- net defined by OOA [i] based on linear OOA(397, 729, F3, 19, 19) (dual of [(729, 19), 13754, 20]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(107−19, 107, 3372)-Net over F3 — Digital
Digital (88, 107, 3372)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3107, 3372, F3, 19) (dual of [3372, 3265, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3107, 6598, F3, 19) (dual of [6598, 6491, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(365, 6562, F3, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3107, 6598, F3, 19) (dual of [6598, 6491, 20]-code), using
(107−19, 107, 863267)-Net in Base 3 — Upper bound on s
There is no (88, 107, 863268)-net in base 3, because
- 1 times m-reduction [i] would yield (88, 106, 863268)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 375 712977 962204 111280 704700 740822 187959 939545 943881 > 3106 [i]