Best Known (247−45, 247, s)-Nets in Base 2
(247−45, 247, 320)-Net over F2 — Constructive and digital
Digital (202, 247, 320)-net over F2, using
- 22 times duplication [i] based on digital (200, 245, 320)-net over F2, using
- t-expansion [i] based on digital (199, 245, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 49, 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, 49, 64)-net over F32, using
- t-expansion [i] based on digital (199, 245, 320)-net over F2, using
(247−45, 247, 882)-Net over F2 — Digital
Digital (202, 247, 882)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 882, F2, 2, 45) (dual of [(882, 2), 1517, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2247, 1031, F2, 2, 45) (dual of [(1031, 2), 1815, 46]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2246, 1031, F2, 2, 45) (dual of [(1031, 2), 1816, 46]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2244, 1030, F2, 2, 45) (dual of [(1030, 2), 1816, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2244, 2060, F2, 45) (dual of [2060, 1816, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(42) [i] based on
- 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(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(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(44) ⊂ Ce(42) [i] based on
- OOA 2-folding [i] based on linear OA(2244, 2060, F2, 45) (dual of [2060, 1816, 46]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2244, 1030, F2, 2, 45) (dual of [(1030, 2), 1816, 46]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2246, 1031, F2, 2, 45) (dual of [(1031, 2), 1816, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2247, 1031, F2, 2, 45) (dual of [(1031, 2), 1815, 46]-NRT-code), using
(247−45, 247, 21001)-Net in Base 2 — Upper bound on s
There is no (202, 247, 21002)-net in base 2, because
- 1 times m-reduction [i] would yield (202, 246, 21002)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 113 169983 816314 149188 462846 373171 436145 414080 940523 316426 303800 058109 374848 > 2246 [i]