Best Known (243−45, 243, s)-Nets in Base 2
(243−45, 243, 320)-Net over F2 — Constructive and digital
Digital (198, 243, 320)-net over F2, using
- 23 times duplication [i] based on digital (195, 240, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 48, 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, 48, 64)-net over F32, using
(243−45, 243, 822)-Net over F2 — Digital
Digital (198, 243, 822)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2243, 822, F2, 2, 45) (dual of [(822, 2), 1401, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2243, 1024, F2, 2, 45) (dual of [(1024, 2), 1805, 46]-NRT-code), using
- OOA 2-folding [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]
- OOA 2-folding [i] based on linear OA(2243, 2048, F2, 45) (dual of [2048, 1805, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(2243, 1024, F2, 2, 45) (dual of [(1024, 2), 1805, 46]-NRT-code), using
(243−45, 243, 18510)-Net in Base 2 — Upper bound on s
There is no (198, 243, 18511)-net in base 2, because
- 1 times m-reduction [i] would yield (198, 242, 18511)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 7 070296 305770 974361 255006 449274 742972 019834 473082 995203 189435 973063 283996 > 2242 [i]