Best Known (44, 44+15, s)-Nets in Base 27
(44, 44+15, 75922)-Net over F27 — Constructive and digital
Digital (44, 59, 75922)-net over F27, using
- net defined by OOA [i] based on linear OOA(2759, 75922, F27, 15, 15) (dual of [(75922, 15), 1138771, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-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(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
(44, 44+15, 530289)-Net over F27 — Digital
Digital (44, 59, 530289)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2759, 530289, F27, 15) (dual of [530289, 530230, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
(44, 44+15, large)-Net in Base 27 — Upper bound on s
There is no (44, 59, large)-net in base 27, because
- 13 times m-reduction [i] would yield (44, 46, large)-net in base 27, but