Best Known (168, 198, s)-Nets in Base 2
(168, 198, 624)-Net over F2 — Constructive and digital
Digital (168, 198, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 33, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
(168, 198, 2051)-Net over F2 — Digital
Digital (168, 198, 2051)-net over F2, using
- 22 times duplication [i] based on digital (166, 196, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2196, 2051, F2, 4, 30) (dual of [(2051, 4), 8008, 31]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2196, 8204, F2, 30) (dual of [8204, 8008, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2196, 8205, F2, 30) (dual of [8205, 8009, 31]-code), using
- 1 times truncation [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2196, 8205, F2, 30) (dual of [8205, 8009, 31]-code), using
- OOA 4-folding [i] based on linear OA(2196, 8204, F2, 30) (dual of [8204, 8008, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2196, 2051, F2, 4, 30) (dual of [(2051, 4), 8008, 31]-NRT-code), using
(168, 198, 60423)-Net in Base 2 — Upper bound on s
There is no (168, 198, 60424)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 401803 300729 438492 484982 054768 903782 789064 370012 189161 890176 > 2198 [i]