Best Known (209, 256, s)-Nets in Base 2
(209, 256, 320)-Net over F2 — Constructive and digital
Digital (209, 256, 320)-net over F2, using
- 21 times duplication [i] based on digital (208, 255, 320)-net over F2, using
- t-expansion [i] based on digital (207, 255, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 51, 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, 51, 64)-net over F32, using
- t-expansion [i] based on digital (207, 255, 320)-net over F2, using
(209, 256, 881)-Net over F2 — Digital
Digital (209, 256, 881)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 881, F2, 2, 47) (dual of [(881, 2), 1506, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2256, 1030, F2, 2, 47) (dual of [(1030, 2), 1804, 48]-NRT-code), using
- 21 times duplication [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
- 21 times duplication [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(2256, 1030, F2, 2, 47) (dual of [(1030, 2), 1804, 48]-NRT-code), using
(209, 256, 20475)-Net in Base 2 — Upper bound on s
There is no (209, 256, 20476)-net in base 2, because
- 1 times m-reduction [i] would yield (209, 255, 20476)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 57903 289215 490661 780330 949444 512441 097627 939236 786364 273095 863184 425448 023678 > 2255 [i]