Best Known (163, 163+41, s)-Nets in Base 2
(163, 163+41, 260)-Net over F2 — Constructive and digital
Digital (163, 204, 260)-net over F2, using
- t-expansion [i] based on digital (162, 204, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 52, 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, 52, 65)-net over F16, using
- 4 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
(163, 163+41, 531)-Net over F2 — Digital
Digital (163, 204, 531)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2204, 531, F2, 2, 41) (dual of [(531, 2), 858, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2204, 1062, F2, 41) (dual of [1062, 858, 42]-code), using
- adding a parity check bit [i] based on linear OA(2203, 1061, F2, 40) (dual of [1061, 858, 41]-code), using
- construction XX applied to C1 = C([1019,34]), C2 = C([1,36]), C3 = C1 + C2 = C([1,34]), and C∩ = C1 ∩ C2 = C([1019,36]) [i] based on
- 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(2175, 1023, F2, 36) (dual of [1023, 848, 37]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2196, 1023, F2, 41) (dual of [1023, 827, 42]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,36}, and designed minimum distance d ≥ |I|+1 = 42 [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(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, 11, F2, 1) (dual of [11, 10, 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([1019,34]), C2 = C([1,36]), C3 = C1 + C2 = C([1,34]), and C∩ = C1 ∩ C2 = C([1019,36]) [i] based on
- adding a parity check bit [i] based on linear OA(2203, 1061, F2, 40) (dual of [1061, 858, 41]-code), using
- OOA 2-folding [i] based on linear OA(2204, 1062, F2, 41) (dual of [1062, 858, 42]-code), using
(163, 163+41, 9405)-Net in Base 2 — Upper bound on s
There is no (163, 204, 9406)-net in base 2, because
- 1 times m-reduction [i] would yield (163, 203, 9406)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12 856866 572406 361715 976815 006170 072477 081014 837335 454738 316043 > 2203 [i]