Best Known (19, 26, s)-Nets in Base 27
(19, 26, 177150)-Net over F27 — Constructive and digital
Digital (19, 26, 177150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 177150, F27, 7, 7) (dual of [(177150, 7), 1240024, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2726, 531451, F27, 7) (dual of [531451, 531425, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(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(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([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2726, 531451, F27, 7) (dual of [531451, 531425, 8]-code), using
(19, 26, 531452)-Net over F27 — Digital
Digital (19, 26, 531452)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2726, 531452, F27, 7) (dual of [531452, 531426, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(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, 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([0,3]) ⊂ C([0,2]) [i] based on
(19, 26, large)-Net in Base 27 — Upper bound on s
There is no (19, 26, large)-net in base 27, because
- 5 times m-reduction [i] would yield (19, 21, large)-net in base 27, but