Best Known (89−12, 89, s)-Nets in Base 27
(89−12, 89, 2796566)-Net over F27 — Constructive and digital
Digital (77, 89, 2796566)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 7, 366)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 366, F27, 4, 4) (dual of [(366, 4), 1457, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(277, 732, F27, 4) (dual of [732, 725, 5]-code), using
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(275, 728, F27, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(275, 728, F27, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(277, 728, F27, 4) (dual of [728, 721, 5]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(273, 728, F27, 2) (dual of [728, 725, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 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
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- OA 2-folding and stacking [i] based on linear OA(277, 732, F27, 4) (dual of [732, 725, 5]-code), using
- net defined by OOA [i] based on linear OOA(277, 366, F27, 4, 4) (dual of [(366, 4), 1457, 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 (3, 7, 366)-net over F27, using
(89−12, 89, large)-Net over F27 — Digital
Digital (77, 89, large)-net over F27, using
- t-expansion [i] based on digital (76, 89, large)-net over F27, using
- 7 times m-reduction [i] based on digital (76, 96, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2796, large, F27, 20) (dual of [large, large−96, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2796, large, F27, 20) (dual of [large, large−96, 21]-code), using
- 7 times m-reduction [i] based on digital (76, 96, large)-net over F27, using
(89−12, 89, large)-Net in Base 27 — Upper bound on s
There is no (77, 89, large)-net in base 27, because
- 10 times m-reduction [i] would yield (77, 79, large)-net in base 27, but