Best Known (81, 93, s)-Nets in Base 27
(81, 93, 2806045)-Net over F27 — Constructive and digital
Digital (81, 93, 2806045)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 11, 9845)-net over F27, using
- net defined by OOA [i] based on linear OOA(2711, 9845, F27, 4, 4) (dual of [(9845, 4), 39369, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2711, 19690, F27, 4) (dual of [19690, 19679, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(274, 19683, F27, 2) (dual of [19683, 19679, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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 Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2711, 19690, F27, 4) (dual of [19690, 19679, 5]-code), using
- net defined by OOA [i] based on linear OOA(2711, 9845, F27, 4, 4) (dual of [(9845, 4), 39369, 5]-NRT-code), using
- digital (20, 26, 1398100)-net over F27, using
- s-reduction based on digital (20, 26, 2796201)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- s-reduction based on digital (20, 26, 2796201)-net over F27, using
- digital (44, 56, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2756, 1398100, F27, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2756, 8388600, F27, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2756, 8388600, F27, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(2756, 1398100, F27, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (7, 11, 9845)-net over F27, using
(81, 93, large)-Net over F27 — Digital
Digital (81, 93, large)-net over F27, using
- t-expansion [i] based on digital (80, 93, large)-net over F27, using
- 8 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
- 8 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
(81, 93, large)-Net in Base 27 — Upper bound on s
There is no (81, 93, large)-net in base 27, because
- 10 times m-reduction [i] would yield (81, 83, large)-net in base 27, but