Best Known (69, 86, s)-Nets in Base 2
(69, 86, 180)-Net over F2 — Constructive and digital
Digital (69, 86, 180)-net over F2, using
- 22 times duplication [i] based on digital (67, 84, 180)-net over F2, using
- trace code for nets [i] based on digital (4, 21, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 21, 45)-net over F16, using
(69, 86, 367)-Net over F2 — Digital
Digital (69, 86, 367)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(286, 367, F2, 2, 17) (dual of [(367, 2), 648, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(286, 524, F2, 2, 17) (dual of [(524, 2), 962, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(285, 524, F2, 2, 17) (dual of [(524, 2), 963, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(285, 1048, F2, 17) (dual of [1048, 963, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(284, 1047, F2, 17) (dual of [1047, 963, 18]-code), using
- adding a parity check bit [i] based on linear OA(283, 1046, F2, 16) (dual of [1046, 963, 17]-code), using
- construction XX applied to C1 = C([1021,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), and C∩ = C1 ∩ C2 = C([1021,14]) [i] based on
- linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(270, 1023, F2, 14) (dual of [1023, 953, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(281, 1023, F2, 17) (dual of [1023, 942, 18]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(260, 1023, F2, 12) (dual of [1023, 963, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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([1021,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), and C∩ = C1 ∩ C2 = C([1021,14]) [i] based on
- adding a parity check bit [i] based on linear OA(283, 1046, F2, 16) (dual of [1046, 963, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(284, 1047, F2, 17) (dual of [1047, 963, 18]-code), using
- OOA 2-folding [i] based on linear OA(285, 1048, F2, 17) (dual of [1048, 963, 18]-code), using
- 21 times duplication [i] based on linear OOA(285, 524, F2, 2, 17) (dual of [(524, 2), 963, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(286, 524, F2, 2, 17) (dual of [(524, 2), 962, 18]-NRT-code), using
(69, 86, 5933)-Net in Base 2 — Upper bound on s
There is no (69, 86, 5934)-net in base 2, because
- 1 times m-reduction [i] would yield (69, 85, 5934)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 38 724051 370715 070805 242859 > 285 [i]