Best Known (44, 44+22, s)-Nets in Base 27
(44, 44+22, 1790)-Net over F27 — Constructive and digital
Digital (44, 66, 1790)-net over F27, using
- 271 times duplication [i] based on digital (43, 65, 1790)-net over F27, using
- net defined by OOA [i] based on linear OOA(2765, 1790, F27, 22, 22) (dual of [(1790, 22), 39315, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2765, 19690, F27, 22) (dual of [19690, 19625, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(21) ⊂ Ce(19) [i] based on
- OA 11-folding and stacking [i] based on linear OA(2765, 19690, F27, 22) (dual of [19690, 19625, 23]-code), using
- net defined by OOA [i] based on linear OOA(2765, 1790, F27, 22, 22) (dual of [(1790, 22), 39315, 23]-NRT-code), using
(44, 44+22, 14321)-Net over F27 — Digital
Digital (44, 66, 14321)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2766, 14321, F27, 22) (dual of [14321, 14255, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, 19694, F27, 22) (dual of [19694, 19628, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2755, 19683, F27, 19) (dual of [19683, 19628, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2766, 19694, F27, 22) (dual of [19694, 19628, 23]-code), using
(44, 44+22, large)-Net in Base 27 — Upper bound on s
There is no (44, 66, large)-net in base 27, because
- 20 times m-reduction [i] would yield (44, 46, large)-net in base 27, but