Best Known (22, 22+23, s)-Nets in Base 81
(22, 22+23, 596)-Net over F81 — Constructive and digital
Digital (22, 45, 596)-net over F81, using
- net defined by OOA [i] based on linear OOA(8145, 596, F81, 23, 23) (dual of [(596, 23), 13663, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 6557, F81, 23) (dual of [6557, 6512, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 6557, F81, 23) (dual of [6557, 6512, 24]-code), using
(22, 22+23, 1640)-Net over F81 — Digital
Digital (22, 45, 1640)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8145, 1640, F81, 4, 23) (dual of [(1640, 4), 6515, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(8145, 6560, F81, 23) (dual of [6560, 6515, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- OOA 4-folding [i] based on linear OA(8145, 6560, F81, 23) (dual of [6560, 6515, 24]-code), using
(22, 22+23, 2641577)-Net in Base 81 — Upper bound on s
There is no (22, 45, 2641578)-net in base 81, because
- 1 times m-reduction [i] would yield (22, 44, 2641578)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 940462 013797 723837 105390 517165 748163 849305 331127 969010 047824 645152 189329 113727 500641 > 8144 [i]