Best Known (45−22, 45, s)-Nets in Base 81
(45−22, 45, 597)-Net over F81 — Constructive and digital
Digital (23, 45, 597)-net over F81, using
- net defined by OOA [i] based on linear OOA(8145, 597, F81, 22, 22) (dual of [(597, 22), 13089, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8145, 6567, F81, 22) (dual of [6567, 6522, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 6569, F81, 22) (dual of [6569, 6524, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8145, 6569, F81, 22) (dual of [6569, 6524, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8145, 6567, F81, 22) (dual of [6567, 6522, 23]-code), using
(45−22, 45, 2189)-Net over F81 — Digital
Digital (23, 45, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8145, 2189, F81, 3, 22) (dual of [(2189, 3), 6522, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8145, 6567, F81, 22) (dual of [6567, 6522, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 6569, F81, 22) (dual of [6569, 6524, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8145, 6569, F81, 22) (dual of [6569, 6524, 23]-code), using
- OOA 3-folding [i] based on linear OA(8145, 6567, F81, 22) (dual of [6567, 6522, 23]-code), using
(45−22, 45, 3938785)-Net in Base 81 — Upper bound on s
There is no (23, 45, 3938786)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 76 177481 706930 891284 506124 066100 304027 447156 197597 551531 911201 571237 372484 628267 443681 > 8145 [i]