Best Known (198, 198+41, s)-Nets in Base 2
(198, 198+41, 380)-Net over F2 — Constructive and digital
Digital (198, 239, 380)-net over F2, using
- t-expansion [i] based on digital (197, 239, 380)-net over F2, using
- 1 times m-reduction [i] based on digital (197, 240, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 48, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 48, 76)-net over F32, using
- 1 times m-reduction [i] based on digital (197, 240, 380)-net over F2, using
(198, 198+41, 1055)-Net over F2 — Digital
Digital (198, 239, 1055)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 1055, F2, 2, 41) (dual of [(1055, 2), 1871, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2239, 2110, F2, 41) (dual of [2110, 1871, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 2111, F2, 41) (dual of [2111, 1872, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(2221, 2049, F2, 41) (dual of [2049, 1828, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2177, 2049, F2, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(218, 62, F2, 7) (dual of [62, 44, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(218, 64, F2, 7) (dual of [64, 46, 8]-code), using
- adding a parity check bit [i] based on linear OA(217, 63, F2, 6) (dual of [63, 46, 7]-code), using
- a “cy†code from Brouwer’s database [i]
- adding a parity check bit [i] based on linear OA(217, 63, F2, 6) (dual of [63, 46, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(218, 64, F2, 7) (dual of [64, 46, 8]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2239, 2111, F2, 41) (dual of [2111, 1872, 42]-code), using
- OOA 2-folding [i] based on linear OA(2239, 2110, F2, 41) (dual of [2110, 1871, 42]-code), using
(198, 198+41, 31707)-Net in Base 2 — Upper bound on s
There is no (198, 239, 31708)-net in base 2, because
- 1 times m-reduction [i] would yield (198, 238, 31708)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 441902 280119 108606 463102 870365 763233 315182 553987 368910 939605 633064 319398 > 2238 [i]