Best Known (42, 42+15, s)-Nets in Base 27
(42, 42+15, 75920)-Net over F27 — Constructive and digital
Digital (42, 57, 75920)-net over F27, using
- net defined by OOA [i] based on linear OOA(2757, 75920, F27, 15, 15) (dual of [(75920, 15), 1138743, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using
(42, 42+15, 319374)-Net over F27 — Digital
Digital (42, 57, 319374)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2757, 319374, F27, 15) (dual of [319374, 319317, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using
(42, 42+15, large)-Net in Base 27 — Upper bound on s
There is no (42, 57, large)-net in base 27, because
- 13 times m-reduction [i] would yield (42, 44, large)-net in base 27, but