Best Known (70, 70+19, s)-Nets in Base 27
(70, 70+19, 59135)-Net over F27 — Constructive and digital
Digital (70, 89, 59135)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 86)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (2, 11, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- digital (1, 5, 38)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (54, 73, 59049)-net over F27, using
- net defined by OOA [i] based on linear OOA(2773, 59049, F27, 19, 19) (dual of [(59049, 19), 1121858, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, 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(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using
- net defined by OOA [i] based on linear OOA(2773, 59049, F27, 19, 19) (dual of [(59049, 19), 1121858, 20]-NRT-code), using
- digital (7, 16, 86)-net over F27, using
(70, 70+19, 59166)-Net in Base 27 — Constructive
(70, 89, 59166)-net in base 27, using
- (u, u+v)-construction [i] based on
- (6, 15, 116)-net in base 27, using
- 1 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
- base change [i] based on digital (2, 12, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 12, 116)-net over F81, using
- 1 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
- digital (55, 74, 59050)-net over F27, using
- net defined by OOA [i] based on linear OOA(2774, 59050, F27, 19, 19) (dual of [(59050, 19), 1121876, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2774, 531451, F27, 19) (dual of [531451, 531377, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2765, 531442, F27, 17) (dual of [531442, 531377, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2774, 531451, F27, 19) (dual of [531451, 531377, 20]-code), using
- net defined by OOA [i] based on linear OOA(2774, 59050, F27, 19, 19) (dual of [(59050, 19), 1121876, 20]-NRT-code), using
- (6, 15, 116)-net in base 27, using
(70, 70+19, 3471015)-Net over F27 — Digital
Digital (70, 89, 3471015)-net over F27, using
(70, 70+19, large)-Net in Base 27 — Upper bound on s
There is no (70, 89, large)-net in base 27, because
- 17 times m-reduction [i] would yield (70, 72, large)-net in base 27, but