Best Known (104−17, 104, s)-Nets in Base 27
(104−17, 104, 1053497)-Net over F27 — Constructive and digital
Digital (87, 104, 1053497)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (15, 23, 4922)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 4922, F27, 8, 8) (dual of [(4922, 8), 39353, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2723, 19688, F27, 8) (dual of [19688, 19665, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2723, 19690, F27, 8) (dual of [19690, 19667, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2716, 19683, F27, 6) (dual of [19683, 19667, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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 Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2723, 19690, F27, 8) (dual of [19690, 19667, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2723, 19688, F27, 8) (dual of [19688, 19665, 9]-code), using
- net defined by OOA [i] based on linear OOA(2723, 4922, F27, 8, 8) (dual of [(4922, 8), 39353, 9]-NRT-code), using
- digital (64, 81, 1048575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (15, 23, 4922)-net over F27, using
(104−17, 104, large)-Net over F27 — Digital
Digital (87, 104, large)-net over F27, using
- t-expansion [i] based on digital (84, 104, large)-net over F27, using
- 2 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- 2 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
(104−17, 104, large)-Net in Base 27 — Upper bound on s
There is no (87, 104, large)-net in base 27, because
- 15 times m-reduction [i] would yield (87, 89, large)-net in base 27, but