Best Known (163, 198, s)-Nets in Base 2
(163, 198, 320)-Net over F2 — Constructive and digital
Digital (163, 198, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (163, 200, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 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, 40, 64)-net over F32, using
(163, 198, 847)-Net over F2 — Digital
Digital (163, 198, 847)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2198, 847, F2, 2, 35) (dual of [(847, 2), 1496, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 1040, F2, 2, 35) (dual of [(1040, 2), 1882, 36]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2194, 1038, F2, 2, 35) (dual of [(1038, 2), 1882, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- 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(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(34) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2194, 1038, F2, 2, 35) (dual of [(1038, 2), 1882, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 1040, F2, 2, 35) (dual of [(1040, 2), 1882, 36]-NRT-code), using
(163, 198, 22072)-Net in Base 2 — Upper bound on s
There is no (163, 198, 22073)-net in base 2, because
- 1 times m-reduction [i] would yield (163, 197, 22073)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 200875 823804 742883 075774 838023 010354 536968 336323 999942 775258 > 2197 [i]