Best Known (45−12, 45, s)-Nets in Base 27
(45−12, 45, 88574)-Net over F27 — Constructive and digital
Digital (33, 45, 88574)-net over F27, using
- net defined by OOA [i] based on linear OOA(2745, 88574, F27, 12, 12) (dual of [(88574, 12), 1062843, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2745, 531444, F27, 12) (dual of [531444, 531399, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2745, 531445, F27, 12) (dual of [531445, 531400, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(2745, 531445, F27, 12) (dual of [531445, 531400, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2745, 531444, F27, 12) (dual of [531444, 531399, 13]-code), using
(45−12, 45, 345938)-Net over F27 — Digital
Digital (33, 45, 345938)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2745, 345938, F27, 12) (dual of [345938, 345893, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using
(45−12, 45, large)-Net in Base 27 — Upper bound on s
There is no (33, 45, large)-net in base 27, because
- 10 times m-reduction [i] would yield (33, 35, large)-net in base 27, but