Best Known (64, 64+34, s)-Nets in Base 27
(64, 64+34, 1158)-Net over F27 — Constructive and digital
Digital (64, 98, 1158)-net over F27, using
- 271 times duplication [i] based on digital (63, 97, 1158)-net over F27, using
- net defined by OOA [i] based on linear OOA(2797, 1158, F27, 34, 34) (dual of [(1158, 34), 39275, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(2797, 19686, F27, 34) (dual of [19686, 19589, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(2797, 19683, F27, 34) (dual of [19683, 19586, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2794, 19683, F27, 33) (dual of [19683, 19589, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 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 X applied to Ce(33) ⊂ Ce(32) [i] based on
- OA 17-folding and stacking [i] based on linear OA(2797, 19686, F27, 34) (dual of [19686, 19589, 35]-code), using
- net defined by OOA [i] based on linear OOA(2797, 1158, F27, 34, 34) (dual of [(1158, 34), 39275, 35]-NRT-code), using
(64, 64+34, 10717)-Net over F27 — Digital
Digital (64, 98, 10717)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2798, 10717, F27, 34) (dual of [10717, 10619, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2798, 19690, F27, 34) (dual of [19690, 19592, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(2797, 19683, F27, 34) (dual of [19683, 19586, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2791, 19683, F27, 32) (dual of [19683, 19592, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(2798, 19690, F27, 34) (dual of [19690, 19592, 35]-code), using
(64, 64+34, large)-Net in Base 27 — Upper bound on s
There is no (64, 98, large)-net in base 27, because
- 32 times m-reduction [i] would yield (64, 66, large)-net in base 27, but