Best Known (86, 103, s)-Nets in Base 27
(86, 103, 1053496)-Net over F27 — Constructive and digital
Digital (86, 103, 1053496)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (14, 22, 4921)-net over F27, using
- net defined by OOA [i] based on linear OOA(2722, 4921, F27, 8, 8) (dual of [(4921, 8), 39346, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2722, 19684, F27, 8) (dual of [19684, 19662, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2722, 19686, F27, 8) (dual of [19686, 19664, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [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(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(2722, 19686, F27, 8) (dual of [19686, 19664, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2722, 19684, F27, 8) (dual of [19684, 19662, 9]-code), using
- net defined by OOA [i] based on linear OOA(2722, 4921, F27, 8, 8) (dual of [(4921, 8), 39346, 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 (14, 22, 4921)-net over F27, using
(86, 103, large)-Net over F27 — Digital
Digital (86, 103, large)-net over F27, using
- t-expansion [i] based on digital (84, 103, large)-net over F27, using
- 3 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
- 3 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
(86, 103, large)-Net in Base 27 — Upper bound on s
There is no (86, 103, large)-net in base 27, because
- 15 times m-reduction [i] would yield (86, 88, large)-net in base 27, but