Best Known (58, 79, s)-Nets in Base 27
(58, 79, 2054)-Net over F27 — Constructive and digital
Digital (58, 79, 2054)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 18, 86)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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, 12, 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, 6, 38)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (40, 61, 1968)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- digital (8, 18, 86)-net over F27, using
(58, 79, 2085)-Net in Base 27 — Constructive
(58, 79, 2085)-net in base 27, using
- 271 times duplication [i] based on (57, 78, 2085)-net in base 27, using
- (u, u+v)-construction [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
- digital (41, 62, 1969)-net over F27, using
- net defined by OOA [i] based on linear OOA(2762, 1969, F27, 21, 21) (dual of [(1969, 21), 41287, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2762, 19691, F27, 21) (dual of [19691, 19629, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(2761, 19684, F27, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2762, 19691, F27, 21) (dual of [19691, 19629, 22]-code), using
- net defined by OOA [i] based on linear OOA(2762, 1969, F27, 21, 21) (dual of [(1969, 21), 41287, 22]-NRT-code), using
- (6, 16, 116)-net in base 27, using
- (u, u+v)-construction [i] based on
(58, 79, 143963)-Net over F27 — Digital
Digital (58, 79, 143963)-net over F27, using
(58, 79, large)-Net in Base 27 — Upper bound on s
There is no (58, 79, large)-net in base 27, because
- 19 times m-reduction [i] would yield (58, 60, large)-net in base 27, but