Best Known (219−39, 219, s)-Nets in Base 2
(219−39, 219, 320)-Net over F2 — Constructive and digital
Digital (180, 219, 320)-net over F2, using
- t-expansion [i] based on digital (179, 219, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (179, 220, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 44, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 44, 64)-net over F32, using
- 1 times m-reduction [i] based on digital (179, 220, 320)-net over F2, using
(219−39, 219, 880)-Net over F2 — Digital
Digital (180, 219, 880)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 880, F2, 2, 39) (dual of [(880, 2), 1541, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 1039, F2, 2, 39) (dual of [(1039, 2), 1859, 40]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2218, 1039, F2, 2, 39) (dual of [(1039, 2), 1860, 40]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2216, 1038, F2, 2, 39) (dual of [(1038, 2), 1860, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2216, 2076, F2, 39) (dual of [2076, 1860, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-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(38) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(2216, 2076, F2, 39) (dual of [2076, 1860, 40]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2216, 1038, F2, 2, 39) (dual of [(1038, 2), 1860, 40]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2218, 1039, F2, 2, 39) (dual of [(1039, 2), 1860, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 1039, F2, 2, 39) (dual of [(1039, 2), 1859, 40]-NRT-code), using
(219−39, 219, 22521)-Net in Base 2 — Upper bound on s
There is no (180, 219, 22522)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 218, 22522)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 421389 750383 706113 338802 588880 700465 298876 831855 514339 274667 931248 > 2218 [i]