Best Known (89, 89+19, s)-Nets in Base 27
(89, 89+19, 932248)-Net over F27 — Constructive and digital
Digital (89, 108, 932248)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 182)-net over F27, using
- net defined by OOA [i] based on linear OOA(2717, 182, F27, 9, 9) (dual of [(182, 9), 1621, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using
- net defined by OOA [i] based on linear OOA(2717, 182, F27, 9, 9) (dual of [(182, 9), 1621, 10]-NRT-code), using
- digital (72, 91, 932066)-net over F27, using
- net defined by OOA [i] based on linear OOA(2791, 932066, F27, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2791, 8388595, F27, 19) (dual of [8388595, 8388504, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2791, 8388595, F27, 19) (dual of [8388595, 8388504, 20]-code), using
- net defined by OOA [i] based on linear OOA(2791, 932066, F27, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- digital (8, 17, 182)-net over F27, using
(89, 89+19, large)-Net over F27 — Digital
Digital (89, 108, large)-net over F27, using
- 272 times duplication [i] based on digital (87, 106, large)-net over F27, using
- t-expansion [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
- t-expansion [i] based on digital (84, 106, large)-net over F27, using
(89, 89+19, large)-Net in Base 27 — Upper bound on s
There is no (89, 108, large)-net in base 27, because
- 17 times m-reduction [i] would yield (89, 91, large)-net in base 27, but