Best Known (60, 60+32, s)-Nets in Base 27
(60, 60+32, 1230)-Net over F27 — Constructive and digital
Digital (60, 92, 1230)-net over F27, using
- 271 times duplication [i] based on digital (59, 91, 1230)-net over F27, using
- net defined by OOA [i] based on linear OOA(2791, 1230, F27, 32, 32) (dual of [(1230, 32), 39269, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(2791, 19680, F27, 32) (dual of [19680, 19589, 33]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(2791, 19683, F27, 32) (dual of [19683, 19592, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(2791, 19680, F27, 32) (dual of [19680, 19589, 33]-code), using
- net defined by OOA [i] based on linear OOA(2791, 1230, F27, 32, 32) (dual of [(1230, 32), 39269, 33]-NRT-code), using
(60, 60+32, 10162)-Net over F27 — Digital
Digital (60, 92, 10162)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2792, 10162, F27, 32) (dual of [10162, 10070, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2792, 19690, F27, 32) (dual of [19690, 19598, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- 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(2785, 19683, F27, 30) (dual of [19683, 19598, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(31) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(2792, 19690, F27, 32) (dual of [19690, 19598, 33]-code), using
(60, 60+32, large)-Net in Base 27 — Upper bound on s
There is no (60, 92, large)-net in base 27, because
- 30 times m-reduction [i] would yield (60, 62, large)-net in base 27, but