Best Known (37, 37+37, s)-Nets in Base 81
(37, 37+37, 730)-Net over F81 — Constructive and digital
Digital (37, 74, 730)-net over F81, using
- t-expansion [i] based on digital (36, 74, 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
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(37, 37+37, 2085)-Net over F81 — Digital
Digital (37, 74, 2085)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8174, 2085, F81, 3, 37) (dual of [(2085, 3), 6181, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 2189, F81, 3, 37) (dual of [(2189, 3), 6493, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8174, 6567, F81, 37) (dual of [6567, 6493, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(8173, 6562, F81, 37) (dual of [6562, 6489, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(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([0,18]) ⊂ C([0,17]) [i] based on
- OOA 3-folding [i] based on linear OA(8174, 6567, F81, 37) (dual of [6567, 6493, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 2189, F81, 3, 37) (dual of [(2189, 3), 6493, 38]-NRT-code), using
(37, 37+37, 5188076)-Net in Base 81 — Upper bound on s
There is no (37, 74, 5188077)-net in base 81, because
- 1 times m-reduction [i] would yield (37, 73, 5188077)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 20 864468 631425 899688 859145 080224 161643 477012 591425 316677 178968 113635 884957 882338 337560 763246 498082 030359 358589 001317 930839 395104 765375 336481 > 8173 [i]