Best Known (259−47, 259, s)-Nets in Base 2
(259−47, 259, 320)-Net over F2 — Constructive and digital
Digital (212, 259, 320)-net over F2, using
- t-expansion [i] based on digital (211, 259, 320)-net over F2, using
- 1 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
- 1 times m-reduction [i] based on digital (211, 260, 320)-net over F2, using
(259−47, 259, 926)-Net over F2 — Digital
Digital (212, 259, 926)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 926, F2, 2, 47) (dual of [(926, 2), 1593, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 1032, F2, 2, 47) (dual of [(1032, 2), 1805, 48]-NRT-code), using
- 2 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
- 2 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
- discarding factors / shortening the dual code based on linear OOA(2259, 1032, F2, 2, 47) (dual of [(1032, 2), 1805, 48]-NRT-code), using
(259−47, 259, 22416)-Net in Base 2 — Upper bound on s
There is no (212, 259, 22417)-net in base 2, because
- 1 times m-reduction [i] would yield (212, 258, 22417)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463357 273351 524780 024104 897691 050940 822229 619738 976708 380681 189322 087500 145280 > 2258 [i]