Best Known (61, 61+12, s)-Nets in Base 27
(61, 61+12, 1404663)-Net over F27 — Constructive and digital
Digital (61, 73, 1404663)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 17, 6563)-net over F27, using
- net defined by OOA [i] based on linear OOA(2717, 6563, F27, 6, 6) (dual of [(6563, 6), 39361, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2717, 19689, F27, 6) (dual of [19689, 19672, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2717, 19690, F27, 6) (dual of [19690, 19673, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [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(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(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(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2717, 19690, F27, 6) (dual of [19690, 19673, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2717, 19689, F27, 6) (dual of [19689, 19672, 7]-code), using
- net defined by OOA [i] based on linear OOA(2717, 6563, F27, 6, 6) (dual of [(6563, 6), 39361, 7]-NRT-code), 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 (11, 17, 6563)-net over F27, using
(61, 61+12, large)-Net over F27 — Digital
Digital (61, 73, large)-net over F27, using
- t-expansion [i] based on digital (60, 73, large)-net over F27, using
- 3 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
- 3 times m-reduction [i] based on digital (60, 76, large)-net over F27, using
(61, 61+12, large)-Net in Base 27 — Upper bound on s
There is no (61, 73, large)-net in base 27, because
- 10 times m-reduction [i] would yield (61, 63, large)-net in base 27, but