Best Known (128, 128+31, s)-Nets in Base 2
(128, 128+31, 260)-Net over F2 — Constructive and digital
Digital (128, 159, 260)-net over F2, using
- t-expansion [i] based on digital (126, 159, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (126, 160, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 40, 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, 40, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (126, 160, 260)-net over F2, using
(128, 128+31, 511)-Net over F2 — Digital
Digital (128, 159, 511)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 511, F2, 2, 31) (dual of [(511, 2), 863, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 531, F2, 2, 31) (dual of [(531, 2), 903, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2159, 1062, F2, 31) (dual of [1062, 903, 32]-code), using
- adding a parity check bit [i] based on linear OA(2158, 1061, F2, 30) (dual of [1061, 903, 31]-code), using
- construction XX applied to C1 = C([1019,24]), C2 = C([1,26]), C3 = C1 + C2 = C([1,24]), and C∩ = C1 ∩ C2 = C([1019,26]) [i] based on
- 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(2130, 1023, F2, 26) (dual of [1023, 893, 27]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2151, 1023, F2, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,26}, and designed minimum distance d ≥ |I|+1 = 32 [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(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([1019,24]), C2 = C([1,26]), C3 = C1 + C2 = C([1,24]), and C∩ = C1 ∩ C2 = C([1019,26]) [i] based on
- adding a parity check bit [i] based on linear OA(2158, 1061, F2, 30) (dual of [1061, 903, 31]-code), using
- OOA 2-folding [i] based on linear OA(2159, 1062, F2, 31) (dual of [1062, 903, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 531, F2, 2, 31) (dual of [(531, 2), 903, 32]-NRT-code), using
(128, 128+31, 9497)-Net in Base 2 — Upper bound on s
There is no (128, 159, 9498)-net in base 2, because
- 1 times m-reduction [i] would yield (128, 158, 9498)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 365627 514106 492560 407208 124726 196333 708546 693176 > 2158 [i]