Best Known (58, 77, s)-Nets in Base 27
(58, 77, 59051)-Net over F27 — Constructive and digital
Digital (58, 77, 59051)-net over F27, using
- 271 times duplication [i] based on digital (57, 76, 59051)-net over F27, using
- net defined by OOA [i] based on linear OOA(2776, 59051, F27, 19, 19) (dual of [(59051, 19), 1121893, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2776, 531460, F27, 19) (dual of [531460, 531384, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2776, 531461, F27, 19) (dual of [531461, 531385, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2757, 531442, F27, 15) (dual of [531442, 531385, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2776, 531461, F27, 19) (dual of [531461, 531385, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2776, 531460, F27, 19) (dual of [531460, 531384, 20]-code), using
- net defined by OOA [i] based on linear OOA(2776, 59051, F27, 19, 19) (dual of [(59051, 19), 1121893, 20]-NRT-code), using
(58, 77, 531465)-Net over F27 — Digital
Digital (58, 77, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2777, 531465, F27, 19) (dual of [531465, 531388, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2773, 531441, F27, 19) (dual of [531441, 531368, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(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(18) ⊂ Ce(13) [i] based on
(58, 77, large)-Net in Base 27 — Upper bound on s
There is no (58, 77, large)-net in base 27, because
- 17 times m-reduction [i] would yield (58, 60, large)-net in base 27, but