Best Known (78−38, 78, s)-Nets in Base 32
(78−38, 78, 224)-Net over F32 — Constructive and digital
Digital (40, 78, 224)-net over F32, using
- 2 times m-reduction [i] based on digital (40, 80, 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, 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 (9, 29, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(78−38, 78, 288)-Net in Base 32 — Constructive
(40, 78, 288)-net in base 32, using
- 30 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
(78−38, 78, 747)-Net over F32 — Digital
Digital (40, 78, 747)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3278, 747, F32, 38) (dual of [747, 669, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3278, 1041, F32, 38) (dual of [1041, 963, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3272, 1024, F32, 38) (dual of [1024, 952, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3261, 1024, F32, 31) (dual of [1024, 963, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(326, 17, F32, 6) (dual of [17, 11, 7]-code or 17-arc in PG(5,32)), using
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- Reed–Solomon code RS(26,32) [i]
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3278, 1041, F32, 38) (dual of [1041, 963, 39]-code), using
(78−38, 78, 386260)-Net in Base 32 — Upper bound on s
There is no (40, 78, 386261)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2521 850084 732726 391206 318392 888410 462100 236099 898354 873033 315629 595525 580084 328943 656171 496776 953528 443350 342459 783708 > 3278 [i]