Best Known (243−44, 243, s)-Nets in Base 2
(243−44, 243, 320)-Net over F2 — Constructive and digital
Digital (199, 243, 320)-net over F2, using
- 2 times m-reduction [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
(243−44, 243, 891)-Net over F2 — Digital
Digital (199, 243, 891)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2243, 891, F2, 2, 44) (dual of [(891, 2), 1539, 45]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2243, 1029, F2, 2, 44) (dual of [(1029, 2), 1815, 45]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2243, 2058, F2, 44) (dual of [2058, 1815, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(2243, 2059, F2, 44) (dual of [2059, 1816, 45]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2244, 2060, F2, 45) (dual of [2060, 1816, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(2243, 2059, F2, 44) (dual of [2059, 1816, 45]-code), using
- OOA 2-folding [i] based on linear OA(2243, 2058, F2, 44) (dual of [2058, 1815, 45]-code), using
- discarding factors / shortening the dual code based on linear OOA(2243, 1029, F2, 2, 44) (dual of [(1029, 2), 1815, 45]-NRT-code), using
(243−44, 243, 19104)-Net in Base 2 — Upper bound on s
There is no (199, 243, 19105)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 14 148008 892118 080562 165547 252715 702494 796472 086473 796954 643300 219962 679696 > 2243 [i]