Best Known (38, 75, s)-Nets in Base 81
(38, 75, 730)-Net over F81 — Constructive and digital
Digital (38, 75, 730)-net over F81, using
- t-expansion [i] based on digital (36, 75, 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
(38, 75, 2189)-Net over F81 — Digital
Digital (38, 75, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8175, 2189, F81, 3, 37) (dual of [(2189, 3), 6492, 38]-NRT-code), using
- 811 times duplication [i] 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
- 811 times duplication [i] based on linear OOA(8174, 2189, F81, 3, 37) (dual of [(2189, 3), 6493, 38]-NRT-code), using
(38, 75, 6622675)-Net in Base 81 — Upper bound on s
There is no (38, 75, 6622676)-net in base 81, because
- 1 times m-reduction [i] would yield (38, 74, 6622676)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 1690 021385 578635 117908 591589 074264 923661 012184 343540 414265 728064 163197 218349 050094 087311 958226 601656 076546 488326 217997 239538 926804 661176 064641 > 8174 [i]