Best Known (84, 84+19, s)-Nets in Base 2
(84, 84+19, 260)-Net over F2 — Constructive and digital
Digital (84, 103, 260)-net over F2, using
- 1 times m-reduction [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
(84, 84+19, 631)-Net over F2 — Digital
Digital (84, 103, 631)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2103, 631, F2, 3, 19) (dual of [(631, 3), 1790, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2103, 687, F2, 3, 19) (dual of [(687, 3), 1958, 20]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2102, 687, F2, 3, 19) (dual of [(687, 3), 1959, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2102, 2061, F2, 19) (dual of [2061, 1959, 20]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(2101, 2060, F2, 19) (dual of [2060, 1959, 20]-code), using
- OOA 3-folding [i] based on linear OA(2102, 2061, F2, 19) (dual of [2061, 1959, 20]-code), using
- 21 times duplication [i] based on linear OOA(2102, 687, F2, 3, 19) (dual of [(687, 3), 1959, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2103, 687, F2, 3, 19) (dual of [(687, 3), 1958, 20]-NRT-code), using
(84, 84+19, 10688)-Net in Base 2 — Upper bound on s
There is no (84, 103, 10689)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 102, 10689)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 074799 840873 674913 926361 771728 > 2102 [i]