Best Known (72−13, 72, s)-Nets in Base 27
(72−13, 72, 1398344)-Net over F27 — Constructive and digital
Digital (59, 72, 1398344)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 244)-net over F27, using
- net defined by OOA [i] based on linear OOA(2711, 244, F27, 6, 6) (dual of [(244, 6), 1453, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2711, 732, F27, 6) (dual of [732, 721, 7]-code), using
- construction XX applied to C1 = C([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [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 = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(279, 728, F27, 5) (dual of [728, 719, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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 = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(277, 728, F27, 4) (dual of [728, 721, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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([727,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([727,4]) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2711, 732, F27, 6) (dual of [732, 721, 7]-code), using
- net defined by OOA [i] based on linear OOA(2711, 244, F27, 6, 6) (dual of [(244, 6), 1453, 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 (5, 11, 244)-net over F27, using
(72−13, 72, large)-Net over F27 — Digital
Digital (59, 72, large)-net over F27, using
- 271 times duplication [i] based on digital (58, 71, large)-net over F27, using
- t-expansion [i] based on digital (56, 71, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- t-expansion [i] based on digital (56, 71, large)-net over F27, using
(72−13, 72, large)-Net in Base 27 — Upper bound on s
There is no (59, 72, large)-net in base 27, because
- 11 times m-reduction [i] would yield (59, 61, large)-net in base 27, but