Best Known (213, 213+47, s)-Nets in Base 2
(213, 213+47, 380)-Net over F2 — Constructive and digital
Digital (213, 260, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 52, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
(213, 213+47, 942)-Net over F2 — Digital
Digital (213, 260, 942)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2260, 942, F2, 2, 47) (dual of [(942, 2), 1624, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 1038, F2, 2, 47) (dual of [(1038, 2), 1816, 48]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2260, 2076, F2, 47) (dual of [2076, 1816, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(42) [i] based on
- linear OA(2254, 2048, F2, 47) (dual of [2048, 1794, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-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
- construction X applied to Ce(46) ⊂ Ce(42) [i] based on
- OOA 2-folding [i] based on linear OA(2260, 2076, F2, 47) (dual of [2076, 1816, 48]-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 1038, F2, 2, 47) (dual of [(1038, 2), 1816, 48]-NRT-code), using
(213, 213+47, 23103)-Net in Base 2 — Upper bound on s
There is no (213, 260, 23104)-net in base 2, because
- 1 times m-reduction [i] would yield (213, 259, 23104)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 926807 444858 312637 143201 802965 864154 072481 207091 010302 832764 854206 191994 446693 > 2259 [i]