Best Known (83−39, 83, s)-Nets in Base 32
(83−39, 83, 240)-Net over F32 — Constructive and digital
Digital (44, 83, 240)-net over F32, using
- 5 times m-reduction [i] based on digital (44, 88, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 33, 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, 55, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 33, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(83−39, 83, 513)-Net in Base 32 — Constructive
(44, 83, 513)-net in base 32, using
- 13 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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, 80, 513)-net over F64, using
(83−39, 83, 1007)-Net over F32 — Digital
Digital (44, 83, 1007)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3283, 1007, F32, 39) (dual of [1007, 924, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3283, 1050, F32, 39) (dual of [1050, 967, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(28) [i] based on
- linear OA(3274, 1024, F32, 39) (dual of [1024, 950, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3257, 1024, F32, 29) (dual of [1024, 967, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(329, 26, F32, 9) (dual of [26, 17, 10]-code or 26-arc in PG(8,32)), using
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- Reed–Solomon code RS(23,32) [i]
- discarding factors / shortening the dual code based on linear OA(329, 32, F32, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
- construction X applied to Ce(38) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3283, 1050, F32, 39) (dual of [1050, 967, 40]-code), using
(83−39, 83, 801233)-Net in Base 32 — Upper bound on s
There is no (44, 83, 801234)-net in base 32, because
- 1 times m-reduction [i] would yield (44, 82, 801234)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2644 262659 469358 444290 279916 533432 231998 376786 237612 588731 684487 347401 662724 421299 378275 017344 392749 656101 139579 550472 039200 > 3282 [i]