Best Known (109, 109+27, s)-Nets in Base 2
(109, 109+27, 260)-Net over F2 — Constructive and digital
Digital (109, 136, 260)-net over F2, using
- t-expansion [i] based on digital (108, 136, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 34, 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, 34, 65)-net over F16, using
(109, 109+27, 436)-Net over F2 — Digital
Digital (109, 136, 436)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2136, 436, F2, 2, 27) (dual of [(436, 2), 736, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2136, 524, F2, 2, 27) (dual of [(524, 2), 912, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2135, 524, F2, 2, 27) (dual of [(524, 2), 913, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2135, 1048, F2, 27) (dual of [1048, 913, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2134, 1047, F2, 27) (dual of [1047, 913, 28]-code), using
- adding a parity check bit [i] based on linear OA(2133, 1046, F2, 26) (dual of [1046, 913, 27]-code), using
- construction XX applied to C1 = C([1021,22]), C2 = C([1,24]), C3 = C1 + C2 = C([1,22]), and C∩ = C1 ∩ C2 = C([1021,24]) [i] based on
- linear OA(2121, 1023, F2, 25) (dual of [1023, 902, 26]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,22}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2120, 1023, F2, 24) (dual of [1023, 903, 25]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 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 = {−2,−1,…,24}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2110, 1023, F2, 22) (dual of [1023, 913, 23]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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
- 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 (see above)
- construction XX applied to C1 = C([1021,22]), C2 = C([1,24]), C3 = C1 + C2 = C([1,22]), and C∩ = C1 ∩ C2 = C([1021,24]) [i] based on
- adding a parity check bit [i] based on linear OA(2133, 1046, F2, 26) (dual of [1046, 913, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2134, 1047, F2, 27) (dual of [1047, 913, 28]-code), using
- OOA 2-folding [i] based on linear OA(2135, 1048, F2, 27) (dual of [1048, 913, 28]-code), using
- 21 times duplication [i] based on linear OOA(2135, 524, F2, 2, 27) (dual of [(524, 2), 913, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2136, 524, F2, 2, 27) (dual of [(524, 2), 912, 28]-NRT-code), using
(109, 109+27, 7557)-Net in Base 2 — Upper bound on s
There is no (109, 136, 7558)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 135, 7558)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 43568 078899 353894 208877 307886 234486 100624 > 2135 [i]