Best Known (86, 86+18, s)-Nets in Base 27
(86, 86+18, 932250)-Net over F27 — Constructive and digital
Digital (86, 104, 932250)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 183)-net over F27, using
- net defined by OOA [i] based on linear OOA(2718, 183, F27, 9, 9) (dual of [(183, 9), 1629, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2718, 733, F27, 9) (dual of [733, 715, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2718, 735, F27, 9) (dual of [735, 717, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2717, 730, F27, 9) (dual of [730, 713, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2718, 735, F27, 9) (dual of [735, 717, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2718, 733, F27, 9) (dual of [733, 715, 10]-code), using
- net defined by OOA [i] based on linear OOA(2718, 183, F27, 9, 9) (dual of [(183, 9), 1629, 10]-NRT-code), using
- digital (68, 86, 932067)-net over F27, using
- net defined by OOA [i] based on linear OOA(2786, 932067, F27, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- net defined by OOA [i] based on linear OOA(2786, 932067, F27, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
- digital (9, 18, 183)-net over F27, using
(86, 86+18, large)-Net over F27 — Digital
Digital (86, 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
(86, 86+18, large)-Net in Base 27 — Upper bound on s
There is no (86, 104, large)-net in base 27, because
- 16 times m-reduction [i] would yield (86, 88, large)-net in base 27, but