Best Known (71, 71+19, s)-Nets in Base 2
(71, 71+19, 152)-Net over F2 — Constructive and digital
Digital (71, 90, 152)-net over F2, using
- 22 times duplication [i] based on digital (69, 88, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 22, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 22, 38)-net over F16, using
(71, 71+19, 273)-Net over F2 — Digital
Digital (71, 90, 273)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(290, 273, F2, 2, 19) (dual of [(273, 2), 456, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(290, 546, F2, 19) (dual of [546, 456, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(290, 547, F2, 19) (dual of [547, 457, 20]-code), using
- adding a parity check bit [i] based on linear OA(289, 546, F2, 18) (dual of [546, 457, 19]-code), using
- construction XX applied to C1 = C([507,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), and C∩ = C1 ∩ C2 = C([507,14]) [i] based on
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,12}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(263, 511, F2, 14) (dual of [511, 448, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−4,−3,…,14}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(254, 511, F2, 12) (dual of [511, 457, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(26, 25, F2, 3) (dual of [25, 19, 4]-code or 25-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, 10, F2, 1) (dual of [10, 9, 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([507,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), and C∩ = C1 ∩ C2 = C([507,14]) [i] based on
- adding a parity check bit [i] based on linear OA(289, 546, F2, 18) (dual of [546, 457, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(290, 547, F2, 19) (dual of [547, 457, 20]-code), using
- OOA 2-folding [i] based on linear OA(290, 546, F2, 19) (dual of [546, 456, 20]-code), using
(71, 71+19, 3918)-Net in Base 2 — Upper bound on s
There is no (71, 90, 3919)-net in base 2, because
- 1 times m-reduction [i] would yield (71, 89, 3919)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 619 069683 225896 529687 624920 > 289 [i]