Best Known (86, 86+18, s)-Nets in Base 2
(86, 86+18, 260)-Net over F2 — Constructive and digital
Digital (86, 104, 260)-net over F2, using
- t-expansion [i] based on digital (84, 104, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 26, 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, 26, 65)-net over F16, using
(86, 86+18, 695)-Net over F2 — Digital
Digital (86, 104, 695)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 695, F2, 2, 18) (dual of [(695, 2), 1286, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2104, 1031, F2, 2, 18) (dual of [(1031, 2), 1958, 19]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2103, 1031, F2, 2, 18) (dual of [(1031, 2), 1959, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2101, 1030, F2, 2, 18) (dual of [(1030, 2), 1959, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2101, 2060, F2, 18) (dual of [2060, 1959, 19]-code), using
- strength reduction [i] based on linear OA(2101, 2060, F2, 19) (dual of [2060, 1959, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2100, 2048, F2, 19) (dual of [2048, 1948, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(289, 2048, F2, 17) (dual of [2048, 1959, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- strength reduction [i] based on linear OA(2101, 2060, F2, 19) (dual of [2060, 1959, 20]-code), using
- OOA 2-folding [i] based on linear OA(2101, 2060, F2, 18) (dual of [2060, 1959, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2101, 1030, F2, 2, 18) (dual of [(1030, 2), 1959, 19]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2103, 1031, F2, 2, 18) (dual of [(1031, 2), 1959, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2104, 1031, F2, 2, 18) (dual of [(1031, 2), 1958, 19]-NRT-code), using
(86, 86+18, 12470)-Net in Base 2 — Upper bound on s
There is no (86, 104, 12471)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 20 296395 579241 859562 802585 954816 > 2104 [i]