Best Known (119, 119+28, s)-Nets in Base 2
(119, 119+28, 260)-Net over F2 — Constructive and digital
Digital (119, 147, 260)-net over F2, using
- t-expansion [i] based on digital (117, 147, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (117, 148, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 37, 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, 37, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (117, 148, 260)-net over F2, using
(119, 119+28, 529)-Net over F2 — Digital
Digital (119, 147, 529)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2147, 529, F2, 2, 28) (dual of [(529, 2), 911, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2147, 1058, F2, 28) (dual of [1058, 911, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2147, 1059, F2, 28) (dual of [1059, 912, 29]-code), using
- 1 times truncation [i] based on linear OA(2148, 1060, F2, 29) (dual of [1060, 912, 30]-code), using
- construction XX applied to C1 = C([1019,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,22}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2121, 1023, F2, 25) (dual of [1023, 902, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2141, 1023, F2, 29) (dual of [1023, 882, 30]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,24}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-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([1019,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- 1 times truncation [i] based on linear OA(2148, 1060, F2, 29) (dual of [1060, 912, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2147, 1059, F2, 28) (dual of [1059, 912, 29]-code), using
- OOA 2-folding [i] based on linear OA(2147, 1058, F2, 28) (dual of [1058, 911, 29]-code), using
(119, 119+28, 8734)-Net in Base 2 — Upper bound on s
There is no (119, 147, 8735)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 178 447275 307174 017322 160686 347925 568987 291943 > 2147 [i]