Best Known (157, 196, s)-Nets in Base 2
(157, 196, 260)-Net over F2 — Constructive and digital
Digital (157, 196, 260)-net over F2, using
- t-expansion [i] based on digital (156, 196, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (156, 200, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 50, 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, 50, 65)-net over F16, using
- 4 times m-reduction [i] based on digital (156, 200, 260)-net over F2, using
(157, 196, 529)-Net over F2 — Digital
Digital (157, 196, 529)-net over F2, using
- 21 times duplication [i] based on digital (156, 195, 529)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2195, 529, F2, 2, 39) (dual of [(529, 2), 863, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2195, 1058, F2, 39) (dual of [1058, 863, 40]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2194, 1057, F2, 39) (dual of [1057, 863, 40]-code), using
- adding a parity check bit [i] based on linear OA(2193, 1056, F2, 38) (dual of [1056, 863, 39]-code), using
- construction XX applied to C1 = C([1019,32]), C2 = C([1,34]), C3 = C1 + C2 = C([1,32]), and C∩ = C1 ∩ C2 = C([1019,34]) [i] based on
- linear OA(2181, 1023, F2, 37) (dual of [1023, 842, 38]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,32}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2165, 1023, F2, 34) (dual of [1023, 858, 35]-code), using 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(2186, 1023, F2, 39) (dual of [1023, 837, 40]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,34}, and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2160, 1023, F2, 32) (dual of [1023, 863, 33]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(26, 27, F2, 3) (dual of [27, 21, 4]-code or 27-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, 6, F2, 1) (dual of [6, 5, 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 C1 = C([1019,32]), C2 = C([1,34]), C3 = C1 + C2 = C([1,32]), and C∩ = C1 ∩ C2 = C([1019,34]) [i] based on
- adding a parity check bit [i] based on linear OA(2193, 1056, F2, 38) (dual of [1056, 863, 39]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2194, 1057, F2, 39) (dual of [1057, 863, 40]-code), using
- OOA 2-folding [i] based on linear OA(2195, 1058, F2, 39) (dual of [1058, 863, 40]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2195, 529, F2, 2, 39) (dual of [(529, 2), 863, 40]-NRT-code), using
(157, 196, 9715)-Net in Base 2 — Upper bound on s
There is no (157, 196, 9716)-net in base 2, because
- 1 times m-reduction [i] would yield (157, 195, 9716)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 50220 685265 040210 394155 691647 891201 470730 186831 392097 547053 > 2195 [i]