Best Known (39, 39+9, s)-Nets in Base 3
(39, 39+9, 1639)-Net over F3 — Constructive and digital
Digital (39, 48, 1639)-net over F3, using
- net defined by OOA [i] based on linear OOA(348, 1639, F3, 9, 9) (dual of [(1639, 9), 14703, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(348, 6557, F3, 9) (dual of [6557, 6509, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(348, 6560, F3, 9) (dual of [6560, 6512, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(348, 6560, F3, 9) (dual of [6560, 6512, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(348, 6557, F3, 9) (dual of [6557, 6509, 10]-code), using
(39, 39+9, 3280)-Net over F3 — Digital
Digital (39, 48, 3280)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(348, 3280, F3, 2, 9) (dual of [(3280, 2), 6512, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(348, 6560, F3, 9) (dual of [6560, 6512, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- OOA 2-folding [i] based on linear OA(348, 6560, F3, 9) (dual of [6560, 6512, 10]-code), using
(39, 39+9, 446883)-Net in Base 3 — Upper bound on s
There is no (39, 48, 446884)-net in base 3, because
- 1 times m-reduction [i] would yield (39, 47, 446884)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 26588 973378 004187 164353 > 347 [i]