Best Known (35, 35+35, s)-Nets in Base 81
(35, 35+35, 486)-Net over F81 — Constructive and digital
Digital (35, 70, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 19, 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, 51, 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, 19, 116)-net over F81, using
(35, 35+35, 2054)-Net over F81 — Digital
Digital (35, 70, 2054)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8170, 2054, F81, 3, 35) (dual of [(2054, 3), 6092, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8170, 2189, F81, 3, 35) (dual of [(2189, 3), 6497, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8170, 6567, F81, 35) (dual of [6567, 6497, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(8169, 6562, F81, 35) (dual of [6562, 6493, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(8165, 6562, F81, 33) (dual of [6562, 6497, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [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 C([0,17]) ⊂ C([0,16]) [i] based on
- OOA 3-folding [i] based on linear OA(8170, 6567, F81, 35) (dual of [6567, 6497, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(8170, 2189, F81, 3, 35) (dual of [(2189, 3), 6497, 36]-NRT-code), using
(35, 35+35, 5001017)-Net in Base 81 — Upper bound on s
There is no (35, 70, 5001018)-net in base 81, because
- 1 times m-reduction [i] would yield (35, 69, 5001018)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 484692 710888 908631 115819 644727 597821 681830 782258 589757 376794 441562 479028 291936 032535 194846 591421 891746 472931 932590 832801 851916 622881 > 8169 [i]