Best Known (258−47, 258, s)-Nets in Base 2
(258−47, 258, 320)-Net over F2 — Constructive and digital
Digital (211, 258, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (211, 260, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 52, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 52, 64)-net over F32, using
(258−47, 258, 911)-Net over F2 — Digital
Digital (211, 258, 911)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 911, F2, 2, 47) (dual of [(911, 2), 1564, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 1031, F2, 2, 47) (dual of [(1031, 2), 1804, 48]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2257, 1031, F2, 2, 47) (dual of [(1031, 2), 1805, 48]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2255, 1030, F2, 2, 47) (dual of [(1030, 2), 1805, 48]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2255, 2060, F2, 47) (dual of [2060, 1805, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(44) [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(2243, 2048, F2, 45) (dual of [2048, 1805, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [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(46) ⊂ Ce(44) [i] based on
- OOA 2-folding [i] based on linear OA(2255, 2060, F2, 47) (dual of [2060, 1805, 48]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2255, 1030, F2, 2, 47) (dual of [(1030, 2), 1805, 48]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2257, 1031, F2, 2, 47) (dual of [(1031, 2), 1805, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 1031, F2, 2, 47) (dual of [(1031, 2), 1804, 48]-NRT-code), using
(258−47, 258, 21750)-Net in Base 2 — Upper bound on s
There is no (211, 258, 21751)-net in base 2, because
- 1 times m-reduction [i] would yield (211, 257, 21751)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 231802 676938 209373 203792 102771 002683 096905 912825 525503 963461 896313 613081 166458 > 2257 [i]