Best Known (22, 22+22, s)-Nets in Base 81
(22, 22+22, 596)-Net over F81 — Constructive and digital
Digital (22, 44, 596)-net over F81, using
- 1 times m-reduction [i] based on 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
- net defined by OOA [i] based on linear OOA(8145, 596, F81, 23, 23) (dual of [(596, 23), 13663, 24]-NRT-code), using
(22, 22+22, 2091)-Net over F81 — Digital
Digital (22, 44, 2091)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8144, 2091, F81, 3, 22) (dual of [(2091, 3), 6229, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8144, 2188, F81, 3, 22) (dual of [(2188, 3), 6520, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8144, 6564, F81, 22) (dual of [6564, 6520, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 6566, F81, 22) (dual of [6566, 6522, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 6566, F81, 22) (dual of [6566, 6522, 23]-code), using
- OOA 3-folding [i] based on linear OA(8144, 6564, F81, 22) (dual of [6564, 6520, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(8144, 2188, F81, 3, 22) (dual of [(2188, 3), 6520, 23]-NRT-code), using
(22, 22+22, 2641577)-Net in Base 81 — Upper bound on s
There is no (22, 44, 2641578)-net in base 81, because
- 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]