Best Known (57−29, 57, s)-Nets in Base 81
(57−29, 57, 468)-Net over F81 — Constructive and digital
Digital (28, 57, 468)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 468, F81, 29, 29) (dual of [(468, 29), 13515, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8157, 6553, F81, 29) (dual of [6553, 6496, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8157, 6553, F81, 29) (dual of [6553, 6496, 30]-code), using
(57−29, 57, 1674)-Net over F81 — Digital
Digital (28, 57, 1674)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8157, 1674, F81, 3, 29) (dual of [(1674, 3), 4965, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8157, 2187, F81, 3, 29) (dual of [(2187, 3), 6504, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 3-folding [i] based on linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(8157, 2187, F81, 3, 29) (dual of [(2187, 3), 6504, 30]-NRT-code), using
(57−29, 57, 3253171)-Net in Base 81 — Upper bound on s
There is no (28, 57, 3253172)-net in base 81, because
- 1 times m-reduction [i] would yield (28, 56, 3253172)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 75017 450565 254656 613912 957588 884559 757283 859468 324502 262099 544863 847595 825187 147355 448237 412174 111088 531841 > 8156 [i]