Best Known (31, 31+28, s)-Nets in Base 81
(31, 31+28, 470)-Net over F81 — Constructive and digital
Digital (31, 59, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (16, 44, 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 (1, 15, 100)-net over F81, using
(31, 31+28, 2845)-Net over F81 — Digital
Digital (31, 59, 2845)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8159, 2845, F81, 2, 28) (dual of [(2845, 2), 5631, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 3287, F81, 2, 28) (dual of [(3287, 2), 6515, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8159, 6574, F81, 28) (dual of [6574, 6515, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 6575, F81, 28) (dual of [6575, 6516, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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(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(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8159, 6575, F81, 28) (dual of [6575, 6516, 29]-code), using
- OOA 2-folding [i] based on linear OA(8159, 6574, F81, 28) (dual of [6574, 6515, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 3287, F81, 2, 28) (dual of [(3287, 2), 6515, 29]-NRT-code), using
(31, 31+28, 8341968)-Net in Base 81 — Upper bound on s
There is no (31, 59, 8341969)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 39867 254953 493305 386620 474432 201220 978586 319573 633972 511082 385882 905150 535524 643525 130836 447328 759795 510648 061281 > 8159 [i]