Best Known (141, 141+35, s)-Nets in Base 2
(141, 141+35, 260)-Net over F2 — Constructive and digital
Digital (141, 176, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (141, 180, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 45, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 45, 65)-net over F16, using
(141, 141+35, 510)-Net over F2 — Digital
Digital (141, 176, 510)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2176, 510, F2, 2, 35) (dual of [(510, 2), 844, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2176, 526, F2, 2, 35) (dual of [(526, 2), 876, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2176, 1052, F2, 35) (dual of [1052, 876, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2151, 1024, F2, 31) (dual of [1024, 873, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2141, 1024, F2, 29) (dual of [1024, 883, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(26, 24, F2, 3) (dual of [24, 18, 4]-code or 24-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
- linear OA(21, 4, F2, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(2176, 1052, F2, 35) (dual of [1052, 876, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2176, 526, F2, 2, 35) (dual of [(526, 2), 876, 36]-NRT-code), using
(141, 141+35, 8986)-Net in Base 2 — Upper bound on s
There is no (141, 176, 8987)-net in base 2, because
- 1 times m-reduction [i] would yield (141, 175, 8987)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 47958 525314 496146 944403 926916 066315 830463 893604 070484 > 2175 [i]