Best Known (60, 60+13, s)-Nets in Base 3
(60, 60+13, 3280)-Net over F3 — Constructive and digital
Digital (60, 73, 3280)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 3280, F3, 13, 13) (dual of [(3280, 13), 42567, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(373, 19681, F3, 13) (dual of [19681, 19608, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(373, 19681, F3, 13) (dual of [19681, 19608, 14]-code), using
(60, 60+13, 6561)-Net over F3 — Digital
Digital (60, 73, 6561)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(373, 6561, F3, 3, 13) (dual of [(6561, 3), 19610, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
(60, 60+13, 795507)-Net in Base 3 — Upper bound on s
There is no (60, 73, 795508)-net in base 3, because
- 1 times m-reduction [i] would yield (60, 72, 795508)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 22528 527271 730987 331681 892208 557945 > 372 [i]