Best Known (40, 40+41, s)-Nets in Base 32
(40, 40+41, 224)-Net over F32 — Constructive and digital
Digital (40, 81, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 29, 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, 52, 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, 29, 104)-net over F32, using
(40, 40+41, 288)-Net in Base 32 — Constructive
(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
(40, 40+41, 590)-Net over F32 — Digital
Digital (40, 81, 590)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3281, 590, F32, 41) (dual of [590, 509, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3281, 1025, F32, 41) (dual of [1025, 944, 42]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3281, 1025, F32, 41) (dual of [1025, 944, 42]-code), using
(40, 40+41, 280885)-Net in Base 32 — Upper bound on s
There is no (40, 81, 280886)-net in base 32, because
- 1 times m-reduction [i] would yield (40, 80, 280886)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 582407 905507 844687 335433 102133 420981 112237 503661 377368 566268 930613 387999 249836 019878 761803 976792 101083 801503 453661 921139 > 3280 [i]