Best Known (126, 126+26, s)-Nets in Base 2
(126, 126+26, 320)-Net over F2 — Constructive and digital
Digital (126, 152, 320)-net over F2, using
- 22 times duplication [i] based on digital (124, 150, 320)-net over F2, using
- t-expansion [i] based on digital (123, 150, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 30, 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, 30, 64)-net over F32, using
- t-expansion [i] based on digital (123, 150, 320)-net over F2, using
(126, 126+26, 834)-Net over F2 — Digital
Digital (126, 152, 834)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2152, 834, F2, 2, 26) (dual of [(834, 2), 1516, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2152, 1039, F2, 2, 26) (dual of [(1039, 2), 1926, 27]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2150, 1038, F2, 2, 26) (dual of [(1038, 2), 1926, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2150, 2076, F2, 26) (dual of [2076, 1926, 27]-code), using
- strength reduction [i] based on linear OA(2150, 2076, F2, 27) (dual of [2076, 1926, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(26) ⊂ Ce(22) [i] based on
- strength reduction [i] based on linear OA(2150, 2076, F2, 27) (dual of [2076, 1926, 28]-code), using
- OOA 2-folding [i] based on linear OA(2150, 2076, F2, 26) (dual of [2076, 1926, 27]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2150, 1038, F2, 2, 26) (dual of [(1038, 2), 1926, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2152, 1039, F2, 2, 26) (dual of [(1039, 2), 1926, 27]-NRT-code), using
(126, 126+26, 18737)-Net in Base 2 — Upper bound on s
There is no (126, 152, 18738)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5712 694355 134577 125446 339399 656328 426494 140924 > 2152 [i]