Best Known (45, 45+44, s)-Nets in Base 32
(45, 45+44, 240)-Net over F32 — Constructive and digital
Digital (45, 89, 240)-net over F32, using
- 2 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
(45, 45+44, 513)-Net in Base 32 — Constructive
(45, 89, 513)-net in base 32, using
- 13 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
(45, 45+44, 740)-Net over F32 — Digital
Digital (45, 89, 740)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3289, 740, F32, 44) (dual of [740, 651, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3289, 1041, F32, 44) (dual of [1041, 952, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(37) [i] based on
- linear OA(3284, 1024, F32, 44) (dual of [1024, 940, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(43) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3289, 1041, F32, 44) (dual of [1041, 952, 45]-code), using
(45, 45+44, 358503)-Net in Base 32 — Upper bound on s
There is no (45, 89, 358504)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 90 856821 276530 606339 012661 407569 709047 366063 350318 932508 736186 842254 651256 796856 445515 359102 695393 899856 486714 282227 453834 192460 790214 > 3289 [i]