Best Known (55, 73, s)-Nets in Base 27
(55, 73, 59051)-Net over F27 — Constructive and digital
Digital (55, 73, 59051)-net over F27, using
- 271 times duplication [i] based on digital (54, 72, 59051)-net over F27, using
- net defined by OOA [i] based on linear OOA(2772, 59051, F27, 18, 18) (dual of [(59051, 18), 1062846, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2772, 531459, F27, 18) (dual of [531459, 531387, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2772, 531460, F27, 18) (dual of [531460, 531388, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2753, 531441, F27, 14) (dual of [531441, 531388, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2772, 531460, F27, 18) (dual of [531460, 531388, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2772, 531459, F27, 18) (dual of [531459, 531387, 19]-code), using
- net defined by OOA [i] based on linear OOA(2772, 59051, F27, 18, 18) (dual of [(59051, 18), 1062846, 19]-NRT-code), using
(55, 73, 531465)-Net over F27 — Digital
Digital (55, 73, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 531465, F27, 18) (dual of [531465, 531392, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(274, 24, F27, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
(55, 73, large)-Net in Base 27 — Upper bound on s
There is no (55, 73, large)-net in base 27, because
- 16 times m-reduction [i] would yield (55, 57, large)-net in base 27, but