Best Known (79−13, 79, s)-Nets in Base 27
(79−13, 79, 1404664)-Net over F27 — Constructive and digital
Digital (66, 79, 1404664)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (12, 18, 6564)-net over F27, using
- net defined by OOA [i] based on linear OOA(2718, 6564, F27, 6, 6) (dual of [(6564, 6), 39366, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2718, 19692, F27, 6) (dual of [19692, 19674, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2718, 19694, F27, 6) (dual of [19694, 19676, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(2716, 19683, F27, 6) (dual of [19683, 19667, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(277, 19683, F27, 3) (dual of [19683, 19676, 4]-code or 19683-cap in PG(6,27)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(2718, 19694, F27, 6) (dual of [19694, 19676, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2718, 19692, F27, 6) (dual of [19692, 19674, 7]-code), using
- net defined by OOA [i] based on linear OOA(2718, 6564, F27, 6, 6) (dual of [(6564, 6), 39366, 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 (12, 18, 6564)-net over F27, using
(79−13, 79, large)-Net over F27 — Digital
Digital (66, 79, large)-net over F27, using
- t-expansion [i] based on digital (64, 79, large)-net over F27, using
- 2 times m-reduction [i] based on digital (64, 81, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- 2 times m-reduction [i] based on digital (64, 81, large)-net over F27, using
(79−13, 79, large)-Net in Base 27 — Upper bound on s
There is no (66, 79, large)-net in base 27, because
- 11 times m-reduction [i] would yield (66, 68, large)-net in base 27, but