Best Known (42, 82, s)-Nets in Base 32
(42, 82, 240)-Net over F32 — Constructive and digital
Digital (42, 82, 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, 51, 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, 82, 513)-Net in Base 32 — Constructive
(42, 82, 513)-net in base 32, using
- 2 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, 82, 766)-Net over F32 — Digital
Digital (42, 82, 766)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 766, F32, 40) (dual of [766, 684, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 1043, F32, 40) (dual of [1043, 961, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- linear OA(3276, 1024, F32, 40) (dual of [1024, 948, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3263, 1024, F32, 33) (dual of [1024, 961, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(326, 19, F32, 6) (dual of [19, 13, 7]-code or 19-arc in PG(5,32)), using
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- Reed–Solomon code RS(26,32) [i]
- discarding factors / shortening the dual code based on linear OA(326, 32, F32, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(3282, 1043, F32, 40) (dual of [1043, 961, 41]-code), using
(42, 82, 397235)-Net in Base 32 — Upper bound on s
There is no (42, 82, 397236)-net in base 32, because
- 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]