Best Known (63−30, 63, s)-Nets in Base 81
(63−30, 63, 486)-Net over F81 — Constructive and digital
Digital (33, 63, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (16, 46, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- digital (2, 17, 116)-net over F81, using
(63−30, 63, 2787)-Net over F81 — Digital
Digital (33, 63, 2787)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8163, 2787, F81, 2, 30) (dual of [(2787, 2), 5511, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8163, 3287, F81, 2, 30) (dual of [(3287, 2), 6511, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8163, 6574, F81, 30) (dual of [6574, 6511, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 6575, F81, 30) (dual of [6575, 6512, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(8163, 6575, F81, 30) (dual of [6575, 6512, 31]-code), using
- OOA 2-folding [i] based on linear OA(8163, 6574, F81, 30) (dual of [6574, 6511, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(8163, 3287, F81, 2, 30) (dual of [(3287, 2), 6511, 31]-NRT-code), using
(63−30, 63, 8323641)-Net in Base 81 — Upper bound on s
There is no (33, 63, 8323642)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 1 716155 801847 639786 071800 192388 506321 910777 950126 546035 508967 660120 061593 679167 820896 784863 040479 354083 346298 462887 026401 > 8163 [i]