Best Known (45−11, 45, s)-Nets in Base 27
(45−11, 45, 106292)-Net over F27 — Constructive and digital
Digital (34, 45, 106292)-net over F27, using
- 271 times duplication [i] based on digital (33, 44, 106292)-net over F27, using
- net defined by OOA [i] based on linear OOA(2744, 106292, F27, 11, 11) (dual of [(106292, 11), 1169168, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2744, 531461, F27, 11) (dual of [531461, 531417, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(2741, 531442, F27, 11) (dual of [531442, 531401, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2725, 531442, F27, 7) (dual of [531442, 531417, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2744, 531461, F27, 11) (dual of [531461, 531417, 12]-code), using
- net defined by OOA [i] based on linear OOA(2744, 106292, F27, 11, 11) (dual of [(106292, 11), 1169168, 12]-NRT-code), using
(45−11, 45, 531465)-Net over F27 — Digital
Digital (34, 45, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2745, 531465, F27, 11) (dual of [531465, 531420, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(2741, 531441, F27, 11) (dual of [531441, 531400, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(10) ⊂ Ce(5) [i] based on
(45−11, 45, large)-Net in Base 27 — Upper bound on s
There is no (34, 45, large)-net in base 27, because
- 9 times m-reduction [i] would yield (34, 36, large)-net in base 27, but