Best Known (137, 137+33, s)-Nets in Base 2
(137, 137+33, 260)-Net over F2 — Constructive and digital
Digital (137, 170, 260)-net over F2, using
- t-expansion [i] based on digital (135, 170, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (135, 172, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 43, 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, 43, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (135, 172, 260)-net over F2, using
(137, 137+33, 531)-Net over F2 — Digital
Digital (137, 170, 531)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2170, 531, F2, 2, 33) (dual of [(531, 2), 892, 34]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2169, 531, F2, 2, 33) (dual of [(531, 2), 893, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2169, 1062, F2, 33) (dual of [1062, 893, 34]-code), using
- adding a parity check bit [i] based on linear OA(2168, 1061, F2, 32) (dual of [1061, 893, 33]-code), using
- construction XX applied to C1 = C([1019,26]), C2 = C([1,28]), C3 = C1 + C2 = C([1,26]), and C∩ = C1 ∩ C2 = C([1019,28]) [i] based on
- 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(2140, 1023, F2, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2161, 1023, F2, 33) (dual of [1023, 862, 34]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,28}, and designed minimum distance d ≥ |I|+1 = 34 [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(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([1019,26]), C2 = C([1,28]), C3 = C1 + C2 = C([1,26]), and C∩ = C1 ∩ C2 = C([1019,28]) [i] based on
- adding a parity check bit [i] based on linear OA(2168, 1061, F2, 32) (dual of [1061, 893, 33]-code), using
- OOA 2-folding [i] based on linear OA(2169, 1062, F2, 33) (dual of [1062, 893, 34]-code), using
- 21 times duplication [i] based on linear OOA(2169, 531, F2, 2, 33) (dual of [(531, 2), 893, 34]-NRT-code), using
(137, 137+33, 10260)-Net in Base 2 — Upper bound on s
There is no (137, 170, 10261)-net in base 2, because
- 1 times m-reduction [i] would yield (137, 169, 10261)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 748 677674 642170 238451 837416 130219 452390 728283 505901 > 2169 [i]