Best Known (18−5, 18, s)-Nets in Base 27
(18−5, 18, 265725)-Net over F27 — Constructive and digital
Digital (13, 18, 265725)-net over F27, using
- net defined by OOA [i] based on linear OOA(2718, 265725, F27, 5, 5) (dual of [(265725, 5), 1328607, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2718, 531451, F27, 5) (dual of [531451, 531433, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2717, 531442, F27, 5) (dual of [531442, 531425, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(279, 531442, F27, 3) (dual of [531442, 531433, 4]-code or 531442-cap in PG(8,27)), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 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 C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2718, 531451, F27, 5) (dual of [531451, 531433, 6]-code), using
(18−5, 18, 531452)-Net over F27 — Digital
Digital (13, 18, 531452)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2718, 531452, F27, 5) (dual of [531452, 531434, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2717, 531442, F27, 5) (dual of [531442, 531425, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(279, 531442, F27, 3) (dual of [531442, 531433, 4]-code or 531442-cap in PG(8,27)), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(279, 10, F27, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,27)), using
- dual of repetition code with length 10 [i]
- linear OA(271, 10, F27, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- Reed–Solomon code RS(26,27) [i]
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
(18−5, 18, large)-Net in Base 27 — Upper bound on s
There is no (13, 18, large)-net in base 27, because
- 3 times m-reduction [i] would yield (13, 15, large)-net in base 27, but