Best Known (75−38, 75, s)-Nets in Base 32
(75−38, 75, 218)-Net over F32 — Constructive and digital
Digital (37, 75, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 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, 49, 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, 26, 98)-net over F32, using
(75−38, 75, 288)-Net in Base 32 — Constructive
(37, 75, 288)-net in base 32, using
- 23 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
(75−38, 75, 556)-Net over F32 — Digital
Digital (37, 75, 556)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3275, 556, F32, 38) (dual of [556, 481, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3275, 1035, F32, 38) (dual of [1035, 960, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [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(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(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3275, 1035, F32, 38) (dual of [1035, 960, 39]-code), using
(75−38, 75, 223468)-Net in Base 32 — Upper bound on s
There is no (37, 75, 223469)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 76959 268858 637812 376582 161156 779527 625775 029475 298597 630415 887293 619022 603298 525940 957949 318113 137114 191272 860536 > 3275 [i]