Best Known (42, 42+41, s)-Nets in Base 32
(42, 42+41, 240)-Net over F32 — Constructive and digital
Digital (42, 83, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 31, 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, 52, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 31, 120)-net over F32, using
(42, 42+41, 513)-Net in Base 32 — Constructive
(42, 83, 513)-net in base 32, using
- 1 times m-reduction [i] based on (42, 84, 513)-net in base 32, using
- base change [i] based on digital (28, 70, 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, 70, 513)-net over F64, using
(42, 42+41, 708)-Net over F32 — Digital
Digital (42, 83, 708)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3283, 708, F32, 41) (dual of [708, 625, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3283, 1041, F32, 41) (dual of [1041, 958, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(3278, 1024, F32, 41) (dual of [1024, 946, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3266, 1024, F32, 35) (dual of [1024, 958, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3283, 1041, F32, 41) (dual of [1041, 958, 42]-code), using
(42, 42+41, 397235)-Net in Base 32 — Upper bound on s
There is no (42, 83, 397236)-net in base 32, because
- 1 times m-reduction [i] would yield (42, 82, 397236)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2644 253509 913949 231902 028057 500919 830011 569338 058436 234749 753115 064150 037908 302755 381078 496506 073199 685331 213448 120174 459384 > 3282 [i]