Best Known (87−42, 87, s)-Nets in Base 32
(87−42, 87, 240)-Net over F32 — Constructive and digital
Digital (45, 87, 240)-net over F32, using
- 4 times m-reduction [i] based on digital (45, 91, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 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 (11, 57, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 34, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(87−42, 87, 513)-Net in Base 32 — Constructive
(45, 87, 513)-net in base 32, using
- 15 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 85, 513)-net over F64, using
(87−42, 87, 857)-Net over F32 — Digital
Digital (45, 87, 857)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3287, 857, F32, 42) (dual of [857, 770, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3287, 1047, F32, 42) (dual of [1047, 960, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- linear OA(3280, 1024, F32, 42) (dual of [1024, 944, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [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(327, 23, F32, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3287, 1047, F32, 42) (dual of [1047, 960, 43]-code), using
(87−42, 87, 481662)-Net in Base 32 — Upper bound on s
There is no (45, 87, 481663)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 88728 439368 439267 079122 172851 808887 347319 068131 142047 049640 899483 041542 168149 174850 371994 327854 765360 312686 886669 501076 625654 141446 > 3287 [i]