Best Known (74−13, 74, s)-Nets in Base 27
(74−13, 74, 1398346)-Net over F27 — Constructive and digital
Digital (61, 74, 1398346)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 246)-net over F27, using
- net defined by OOA [i] based on linear OOA(2713, 246, F27, 6, 6) (dual of [(246, 6), 1463, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2713, 738, F27, 6) (dual of [738, 725, 7]-code), using
- construction XX applied to C1 = C([725,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([725,2]) [i] based on
- linear OA(279, 728, F27, 5) (dual of [728, 719, 6]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,−1,0,1}, and designed minimum distance d ≥ |I|+1 = 6 [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(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,2}, and designed minimum distance d ≥ |I|+1 = 7 [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(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- 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
- construction XX applied to C1 = C([725,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([725,2]) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2713, 738, F27, 6) (dual of [738, 725, 7]-code), using
- net defined by OOA [i] based on linear OOA(2713, 246, F27, 6, 6) (dual of [(246, 6), 1463, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (7, 13, 246)-net over F27, using
(74−13, 74, large)-Net over F27 — Digital
Digital (61, 74, large)-net over F27, using
- t-expansion [i] based on digital (60, 74, large)-net over F27, using
- 2 times m-reduction [i] based on digital (60, 76, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- 2 times m-reduction [i] based on digital (60, 76, large)-net over F27, using
(74−13, 74, large)-Net in Base 27 — Upper bound on s
There is no (61, 74, large)-net in base 27, because
- 11 times m-reduction [i] would yield (61, 63, large)-net in base 27, but