Best Known (23, 23+12, s)-Nets in Base 27
(23, 23+12, 3281)-Net over F27 — Constructive and digital
Digital (23, 35, 3281)-net over F27, using
- 271 times duplication [i] based on digital (22, 34, 3281)-net over F27, using
- net defined by OOA [i] based on linear OOA(2734, 3281, F27, 12, 12) (dual of [(3281, 12), 39338, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2734, 19686, F27, 12) (dual of [19686, 19652, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(2734, 19683, F27, 12) (dual of [19683, 19649, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2731, 19683, F27, 11) (dual of [19683, 19652, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(2734, 19686, F27, 12) (dual of [19686, 19652, 13]-code), using
- net defined by OOA [i] based on linear OOA(2734, 3281, F27, 12, 12) (dual of [(3281, 12), 39338, 13]-NRT-code), using
(23, 23+12, 12808)-Net over F27 — Digital
Digital (23, 35, 12808)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2735, 12808, F27, 12) (dual of [12808, 12773, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2735, 19690, F27, 12) (dual of [19690, 19655, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(2734, 19683, F27, 12) (dual of [19683, 19649, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2728, 19683, F27, 10) (dual of [19683, 19655, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(2735, 19690, F27, 12) (dual of [19690, 19655, 13]-code), using
(23, 23+12, large)-Net in Base 27 — Upper bound on s
There is no (23, 35, large)-net in base 27, because
- 10 times m-reduction [i] would yield (23, 25, large)-net in base 27, but