Best Known (9, 9+9, s)-Nets in Base 27
(9, 9+9, 183)-Net over F27 — Constructive and digital
Digital (9, 18, 183)-net over F27, using
- net defined by OOA [i] based on linear OOA(2718, 183, F27, 9, 9) (dual of [(183, 9), 1629, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2718, 733, F27, 9) (dual of [733, 715, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2718, 735, F27, 9) (dual of [735, 717, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2717, 730, F27, 9) (dual of [730, 713, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2718, 735, F27, 9) (dual of [735, 717, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2718, 733, F27, 9) (dual of [733, 715, 10]-code), using
(9, 9+9, 386)-Net over F27 — Digital
Digital (9, 18, 386)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2718, 386, F27, 9) (dual of [386, 368, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2718, 728, F27, 9) (dual of [728, 710, 10]-code), using
(9, 9+9, 103126)-Net in Base 27 — Upper bound on s
There is no (9, 18, 103127)-net in base 27, because
- 1 times m-reduction [i] would yield (9, 17, 103127)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 2 153771 101845 175021 928473 > 2717 [i]