Best Known (127−13, 127, s)-Nets in Base 3
(127−13, 127, 1398104)-Net over F3 — Constructive and digital
Digital (114, 127, 1398104)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- digital (0, 6, 4)-net over F3, using
(127−13, 127, 2796205)-Net over F3 — Digital
Digital (114, 127, 2796205)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 2796205, F3, 3, 13) (dual of [(2796205, 3), 8388488, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(36, 4, F3, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;6,3) [i]
- linear OOA(3121, 2796201, F3, 3, 13) (dual of [(2796201, 3), 8388482, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- linear OOA(36, 4, F3, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(127−13, 127, large)-Net in Base 3 — Upper bound on s
There is no (114, 127, large)-net in base 3, because
- 11 times m-reduction [i] would yield (114, 116, large)-net in base 3, but