Best Known (67, 67+36, s)-Nets in Base 27
(67, 67+36, 1093)-Net over F27 — Constructive and digital
Digital (67, 103, 1093)-net over F27, using
- net defined by OOA [i] based on linear OOA(27103, 1093, F27, 36, 36) (dual of [(1093, 36), 39245, 37]-NRT-code), using
- OA 18-folding and stacking [i] based on linear OA(27103, 19674, F27, 36) (dual of [19674, 19571, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(27103, 19683, F27, 36) (dual of [19683, 19580, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- discarding factors / shortening the dual code based on linear OA(27103, 19683, F27, 36) (dual of [19683, 19580, 37]-code), using
- OA 18-folding and stacking [i] based on linear OA(27103, 19674, F27, 36) (dual of [19674, 19571, 37]-code), using
(67, 67+36, 10230)-Net over F27 — Digital
Digital (67, 103, 10230)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27103, 10230, F27, 36) (dual of [10230, 10127, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(27103, 19683, F27, 36) (dual of [19683, 19580, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- discarding factors / shortening the dual code based on linear OA(27103, 19683, F27, 36) (dual of [19683, 19580, 37]-code), using
(67, 67+36, large)-Net in Base 27 — Upper bound on s
There is no (67, 103, large)-net in base 27, because
- 34 times m-reduction [i] would yield (67, 69, large)-net in base 27, but