Best Known (38, 38+39, s)-Nets in Base 32
(38, 38+39, 218)-Net over F32 — Constructive and digital
Digital (38, 77, 218)-net over F32, using
- 1 times m-reduction [i] based on digital (38, 78, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 27, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (11, 51, 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 (7, 27, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(38, 38+39, 288)-Net in Base 32 — Constructive
(38, 77, 288)-net in base 32, using
- t-expansion [i] based on (37, 77, 288)-net in base 32, using
- 21 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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, 70, 288)-net over F128, using
- 21 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
(38, 38+39, 567)-Net over F32 — Digital
Digital (38, 77, 567)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3277, 567, F32, 39) (dual of [567, 490, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3277, 1025, F32, 39) (dual of [1025, 948, 40]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3277, 1025, F32, 39) (dual of [1025, 948, 40]-code), using
(38, 38+39, 268187)-Net in Base 32 — Upper bound on s
There is no (38, 77, 268188)-net in base 32, because
- 1 times m-reduction [i] would yield (38, 76, 268188)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 462752 751426 789051 725871 001598 727656 596398 836204 286505 721166 047199 098166 796986 205234 237185 634545 653058 742986 494717 > 3276 [i]