Best Known (140−27, 140, s)-Nets in Base 2
(140−27, 140, 260)-Net over F2 — Constructive and digital
Digital (113, 140, 260)-net over F2, using
- t-expansion [i] based on digital (111, 140, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 35, 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, 35, 65)-net over F16, using
(140−27, 140, 494)-Net over F2 — Digital
Digital (113, 140, 494)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2140, 494, F2, 2, 27) (dual of [(494, 2), 848, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2140, 531, F2, 2, 27) (dual of [(531, 2), 922, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2139, 531, F2, 2, 27) (dual of [(531, 2), 923, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2139, 1062, F2, 27) (dual of [1062, 923, 28]-code), using
- adding a parity check bit [i] based on linear OA(2138, 1061, F2, 26) (dual of [1061, 923, 27]-code), using
- construction XX applied to C1 = C([1019,20]), C2 = C([1,22]), C3 = C1 + C2 = C([1,20]), and C∩ = C1 ∩ C2 = C([1019,22]) [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 = {−4,−3,…,20}, and designed minimum distance d ≥ |I|+1 = 26 [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(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(2100, 1023, F2, 20) (dual of [1023, 923, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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([1019,20]), C2 = C([1,22]), C3 = C1 + C2 = C([1,20]), and C∩ = C1 ∩ C2 = C([1019,22]) [i] based on
- adding a parity check bit [i] based on linear OA(2138, 1061, F2, 26) (dual of [1061, 923, 27]-code), using
- OOA 2-folding [i] based on linear OA(2139, 1062, F2, 27) (dual of [1062, 923, 28]-code), using
- 21 times duplication [i] based on linear OOA(2139, 531, F2, 2, 27) (dual of [(531, 2), 923, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2140, 531, F2, 2, 27) (dual of [(531, 2), 922, 28]-NRT-code), using
(140−27, 140, 9359)-Net in Base 2 — Upper bound on s
There is no (113, 140, 9360)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 139, 9360)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 697821 265042 178468 785538 986291 712684 015943 > 2139 [i]