Best Known (30−10, 30, s)-Nets in Base 27
(30−10, 30, 3938)-Net over F27 — Constructive and digital
Digital (20, 30, 3938)-net over F27, using
- 271 times duplication [i] based on digital (19, 29, 3938)-net over F27, using
- net defined by OOA [i] based on linear OOA(2729, 3938, F27, 10, 10) (dual of [(3938, 10), 39351, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2729, 19690, F27, 10) (dual of [19690, 19661, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(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(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(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(2729, 19690, F27, 10) (dual of [19690, 19661, 11]-code), using
- net defined by OOA [i] based on linear OOA(2729, 3938, F27, 10, 10) (dual of [(3938, 10), 39351, 11]-NRT-code), using
(30−10, 30, 19694)-Net over F27 — Digital
Digital (20, 30, 19694)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2730, 19694, F27, 10) (dual of [19694, 19664, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 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(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
(30−10, 30, large)-Net in Base 27 — Upper bound on s
There is no (20, 30, large)-net in base 27, because
- 8 times m-reduction [i] would yield (20, 22, large)-net in base 27, but