Best Known (84−44, 84, s)-Nets in Base 32
(84−44, 84, 218)-Net over F32 — Constructive and digital
Digital (40, 84, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 29, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (11, 55, 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 (7, 29, 98)-net over F32, using
(84−44, 84, 288)-Net in Base 32 — Constructive
(40, 84, 288)-net in base 32, using
- 24 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
- base change [i] based on (22, 90, 288)-net in base 64, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on (22, 90, 288)-net in base 64, using
(84−44, 84, 513)-Net over F32 — Digital
Digital (40, 84, 513)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3284, 513, F32, 2, 44) (dual of [(513, 2), 942, 45]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3284, 1026, F32, 44) (dual of [1026, 942, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [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(3282, 1024, F32, 43) (dual of [1024, 942, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- OOA 2-folding [i] based on linear OA(3284, 1026, F32, 44) (dual of [1026, 942, 45]-code), using
(84−44, 84, 163078)-Net in Base 32 — Upper bound on s
There is no (40, 84, 163079)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 707723 197611 578225 347164 877323 820365 238582 628222 639748 160480 539647 665271 822591 073244 468584 315959 764560 397720 268659 679401 519984 > 3284 [i]