Best Known (71−35, 71, s)-Nets in Base 81
(71−35, 71, 730)-Net over F81 — Constructive and digital
Digital (36, 71, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
(71−35, 71, 2189)-Net over F81 — Digital
Digital (36, 71, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8171, 2189, F81, 3, 35) (dual of [(2189, 3), 6496, 36]-NRT-code), using
- 811 times duplication [i] 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
- 811 times duplication [i] based on linear OOA(8170, 2189, F81, 3, 35) (dual of [(2189, 3), 6497, 36]-NRT-code), using
(71−35, 71, 6476231)-Net in Base 81 — Upper bound on s
There is no (36, 71, 6476232)-net in base 81, because
- 1 times m-reduction [i] would yield (36, 70, 6476232)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 39 260099 491629 020535 443257 268896 720848 624422 274954 956978 491639 804742 564828 030456 448134 671120 463808 978977 976455 474434 330552 013655 086721 > 8170 [i]