Best Known (72−37, 72, s)-Nets in Base 32
(72−37, 72, 202)-Net over F32 — Constructive and digital
Digital (35, 72, 202)-net over F32, using
- 1 times m-reduction [i] based on digital (35, 73, 202)-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 (9, 47, 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 (7, 26, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(72−37, 72, 288)-Net in Base 32 — Constructive
(35, 72, 288)-net in base 32, using
- 19 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 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, 65, 288)-net over F128, using
(72−37, 72, 516)-Net over F32 — Digital
Digital (35, 72, 516)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3272, 516, F32, 2, 37) (dual of [(516, 2), 960, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3272, 1032, F32, 37) (dual of [1032, 960, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(3270, 1024, F32, 37) (dual of [1024, 954, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(3272, 1032, F32, 37) (dual of [1032, 960, 38]-code), using
(72−37, 72, 210731)-Net in Base 32 — Upper bound on s
There is no (35, 72, 210732)-net in base 32, because
- 1 times m-reduction [i] would yield (35, 71, 210732)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 73393 080114 525649 512307 897688 587224 028481 072381 230826 819170 704787 187749 828995 714129 109857 964819 120404 539230 > 3271 [i]