Best Known (259−43, 259, s)-Nets in Base 2
(259−43, 259, 490)-Net over F2 — Constructive and digital
Digital (216, 259, 490)-net over F2, using
- t-expansion [i] based on digital (215, 259, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (215, 260, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 52, 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
- trace code for nets [i] based on digital (7, 52, 98)-net over F32, using
- 1 times m-reduction [i] based on digital (215, 260, 490)-net over F2, using
(259−43, 259, 1375)-Net over F2 — Digital
Digital (216, 259, 1375)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 1375, F2, 3, 43) (dual of [(1375, 3), 3866, 44]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2259, 4125, F2, 43) (dual of [4125, 3866, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(2259, 4126, F2, 43) (dual of [4126, 3867, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(38) [i] based on
- linear OA(2253, 4096, F2, 43) (dual of [4096, 3843, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2229, 4096, F2, 39) (dual of [4096, 3867, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(42) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(2259, 4126, F2, 43) (dual of [4126, 3867, 44]-code), using
- OOA 3-folding [i] based on linear OA(2259, 4125, F2, 43) (dual of [4125, 3866, 44]-code), using
(259−43, 259, 43306)-Net in Base 2 — Upper bound on s
There is no (216, 259, 43307)-net in base 2, because
- 1 times m-reduction [i] would yield (216, 258, 43307)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463340 359100 031921 231477 145242 864944 865936 389899 336456 897287 885570 444066 064384 > 2258 [i]