Best Known (15, 15+8, s)-Nets in Base 27
(15, 15+8, 4922)-Net over F27 — Constructive and digital
Digital (15, 23, 4922)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 4922, F27, 8, 8) (dual of [(4922, 8), 39353, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2723, 19688, F27, 8) (dual of [19688, 19665, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2723, 19690, F27, 8) (dual of [19690, 19667, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2723, 19690, F27, 8) (dual of [19690, 19667, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2723, 19688, F27, 8) (dual of [19688, 19665, 9]-code), using
(15, 15+8, 19690)-Net over F27 — Digital
Digital (15, 23, 19690)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2723, 19690, F27, 8) (dual of [19690, 19667, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(5) [i] based on
(15, 15+8, large)-Net in Base 27 — Upper bound on s
There is no (15, 23, large)-net in base 27, because
- 6 times m-reduction [i] would yield (15, 17, large)-net in base 27, but