Best Known (41, 74, s)-Nets in Base 81
(41, 74, 730)-Net over F81 — Constructive and digital
Digital (41, 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
(41, 74, 4828)-Net over F81 — Digital
Digital (41, 74, 4828)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8174, 4828, F81, 33) (dual of [4828, 4754, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8174, 6591, F81, 33) (dual of [6591, 6517, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,11]) [i] based on
- 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(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,16]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8174, 6591, F81, 33) (dual of [6591, 6517, 34]-code), using
(41, 74, large)-Net in Base 81 — Upper bound on s
There is no (41, 74, large)-net in base 81, because
- 31 times m-reduction [i] would yield (41, 43, large)-net in base 81, but