Best Known (98−17, 98, s)-Nets in Base 27
(98−17, 98, 1048759)-Net over F27 — Constructive and digital
Digital (81, 98, 1048759)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 17, 184)-net over F27, using
- net defined by OOA [i] based on linear OOA(2717, 184, F27, 8, 8) (dual of [(184, 8), 1455, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2717, 736, F27, 8) (dual of [736, 719, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2717, 737, F27, 8) (dual of [737, 720, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(2715, 729, F27, 8) (dual of [729, 714, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(279, 729, F27, 5) (dual of [729, 720, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-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(2717, 737, F27, 8) (dual of [737, 720, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2717, 736, F27, 8) (dual of [736, 719, 9]-code), using
- net defined by OOA [i] based on linear OOA(2717, 184, F27, 8, 8) (dual of [(184, 8), 1455, 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 (9, 17, 184)-net over F27, using
(98−17, 98, large)-Net over F27 — Digital
Digital (81, 98, large)-net over F27, using
- t-expansion [i] based on digital (80, 98, large)-net over F27, using
- 3 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- 3 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
(98−17, 98, large)-Net in Base 27 — Upper bound on s
There is no (81, 98, large)-net in base 27, because
- 15 times m-reduction [i] would yield (81, 83, large)-net in base 27, but