Best Known (46−22, 46, s)-Nets in Base 81
(46−22, 46, 597)-Net over F81 — Constructive and digital
Digital (24, 46, 597)-net over F81, using
- 1 times m-reduction [i] based on digital (24, 47, 597)-net over F81, using
- net defined by OOA [i] based on linear OOA(8147, 597, F81, 23, 23) (dual of [(597, 23), 13684, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8147, 6568, F81, 23) (dual of [6568, 6521, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 6569, F81, 23) (dual of [6569, 6522, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] 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]
- 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(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(8147, 6569, F81, 23) (dual of [6569, 6522, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8147, 6568, F81, 23) (dual of [6568, 6521, 24]-code), using
- net defined by OOA [i] based on linear OOA(8147, 597, F81, 23, 23) (dual of [(597, 23), 13684, 24]-NRT-code), using
(46−22, 46, 2596)-Net over F81 — Digital
Digital (24, 46, 2596)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8146, 2596, F81, 2, 22) (dual of [(2596, 2), 5146, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 3286, F81, 2, 22) (dual of [(3286, 2), 6526, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8146, 6572, F81, 22) (dual of [6572, 6526, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [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(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(8146, 6572, F81, 22) (dual of [6572, 6526, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 3286, F81, 2, 22) (dual of [(3286, 2), 6526, 23]-NRT-code), using
(46−22, 46, 5873015)-Net in Base 81 — Upper bound on s
There is no (24, 46, 5873016)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 6170 371918 391860 057990 923152 602525 220777 822907 567841 329554 230382 261005 302122 575371 006081 > 8146 [i]