Best Known (81−40, 81, s)-Nets in Base 32
(81−40, 81, 224)-Net over F32 — Constructive and digital
Digital (41, 81, 224)-net over F32, using
- 2 times m-reduction [i] based on digital (41, 83, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 30, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (11, 53, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (9, 30, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(81−40, 81, 288)-Net in Base 32 — Constructive
(41, 81, 288)-net in base 32, using
- t-expansion [i] based on (40, 81, 288)-net in base 32, using
- 27 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
- base change [i] based on (22, 90, 288)-net in base 64, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on (22, 90, 288)-net in base 64, using
- 27 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
(81−40, 81, 697)-Net over F32 — Digital
Digital (41, 81, 697)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3281, 697, F32, 40) (dual of [697, 616, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3281, 1041, F32, 40) (dual of [1041, 960, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3276, 1024, F32, 40) (dual of [1024, 948, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3264, 1024, F32, 34) (dual of [1024, 960, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3281, 1041, F32, 40) (dual of [1041, 960, 41]-code), using
(81−40, 81, 334032)-Net in Base 32 — Upper bound on s
There is no (41, 81, 334033)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 82 634335 689448 601524 172988 818806 901888 606710 020908 695310 462572 137651 576702 041617 529649 474745 558674 280749 734300 283015 069008 > 3281 [i]