Best Known (105−17, 105, s)-Nets in Base 27
(105−17, 105, 1053498)-Net over F27 — Constructive and digital
Digital (88, 105, 1053498)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (16, 24, 4923)-net over F27, using
- net defined by OOA [i] based on linear OOA(2724, 4923, F27, 8, 8) (dual of [(4923, 8), 39360, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2724, 19692, F27, 8) (dual of [19692, 19668, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2724, 19694, F27, 8) (dual of [19694, 19670, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(2713, 19683, F27, 5) (dual of [19683, 19670, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(2724, 19694, F27, 8) (dual of [19694, 19670, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2724, 19692, F27, 8) (dual of [19692, 19668, 9]-code), using
- net defined by OOA [i] based on linear OOA(2724, 4923, F27, 8, 8) (dual of [(4923, 8), 39360, 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 (16, 24, 4923)-net over F27, using
(105−17, 105, large)-Net over F27 — Digital
Digital (88, 105, large)-net over F27, using
- t-expansion [i] based on digital (84, 105, large)-net over F27, using
- 1 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
- 1 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
(105−17, 105, large)-Net in Base 27 — Upper bound on s
There is no (88, 105, large)-net in base 27, because
- 15 times m-reduction [i] would yield (88, 90, large)-net in base 27, but