Best Known (34, 34+34, s)-Nets in Base 81
(34, 34+34, 470)-Net over F81 — Constructive and digital
Digital (34, 68, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 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, 50, 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, 18, 100)-net over F81, using
(34, 34+34, 2040)-Net over F81 — Digital
Digital (34, 68, 2040)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8168, 2040, F81, 3, 34) (dual of [(2040, 3), 6052, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8168, 2188, F81, 3, 34) (dual of [(2188, 3), 6496, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8168, 6564, F81, 34) (dual of [6564, 6496, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(8168, 6566, F81, 34) (dual of [6566, 6498, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(8168, 6566, F81, 34) (dual of [6566, 6498, 35]-code), using
- OOA 3-folding [i] based on linear OA(8168, 6564, F81, 34) (dual of [6564, 6496, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(8168, 2188, F81, 3, 34) (dual of [(2188, 3), 6496, 35]-NRT-code), using
(34, 34+34, 3861840)-Net in Base 81 — Upper bound on s
There is no (34, 68, 3861841)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 5983 862111 232974 397531 166581 232677 658397 698634 819449 865602 319999 983599 009623 634224 730557 720000 814205 421983 563361 715413 355925 358161 > 8168 [i]