Best Known (193−35, 193, s)-Nets in Base 2
(193−35, 193, 320)-Net over F2 — Constructive and digital
Digital (158, 193, 320)-net over F2, using
- 23 times duplication [i] based on digital (155, 190, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 38, 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, 38, 64)-net over F32, using
(193−35, 193, 756)-Net over F2 — Digital
Digital (158, 193, 756)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2193, 756, F2, 2, 35) (dual of [(756, 2), 1319, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 1032, F2, 2, 35) (dual of [(1032, 2), 1871, 36]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2189, 1030, F2, 2, 35) (dual of [(1030, 2), 1871, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2189, 2060, F2, 35) (dual of [2060, 1871, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [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(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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 X applied to Ce(34) ⊂ Ce(32) [i] based on
- OOA 2-folding [i] based on linear OA(2189, 2060, F2, 35) (dual of [2060, 1871, 36]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2189, 1030, F2, 2, 35) (dual of [(1030, 2), 1871, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 1032, F2, 2, 35) (dual of [(1032, 2), 1871, 36]-NRT-code), using
(193−35, 193, 17997)-Net in Base 2 — Upper bound on s
There is no (158, 193, 17998)-net in base 2, because
- 1 times m-reduction [i] would yield (158, 192, 17998)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6280 373044 247675 079009 633497 334669 806431 006009 043146 774428 > 2192 [i]