Best Known (244−41, 244, s)-Nets in Base 2
(244−41, 244, 490)-Net over F2 — Constructive and digital
Digital (203, 244, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (203, 245, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 49, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 49, 98)-net over F32, using
(244−41, 244, 1285)-Net over F2 — Digital
Digital (203, 244, 1285)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 1285, F2, 3, 41) (dual of [(1285, 3), 3611, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 1370, F2, 3, 41) (dual of [(1370, 3), 3866, 42]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2243, 1370, F2, 3, 41) (dual of [(1370, 3), 3867, 42]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2243, 4110, F2, 41) (dual of [4110, 3867, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2242, 4109, F2, 41) (dual of [4109, 3867, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- linear OA(2241, 4096, F2, 41) (dual of [4096, 3855, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2229, 4096, F2, 39) (dual of [4096, 3867, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(40) ⊂ Ce(38) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2242, 4109, F2, 41) (dual of [4109, 3867, 42]-code), using
- OOA 3-folding [i] based on linear OA(2243, 4110, F2, 41) (dual of [4110, 3867, 42]-code), using
- 21 times duplication [i] based on linear OOA(2243, 1370, F2, 3, 41) (dual of [(1370, 3), 3867, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 1370, F2, 3, 41) (dual of [(1370, 3), 3866, 42]-NRT-code), using
(244−41, 244, 37712)-Net in Base 2 — Upper bound on s
There is no (203, 244, 37713)-net in base 2, because
- 1 times m-reduction [i] would yield (203, 243, 37713)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14 141199 206979 510845 650770 001740 070718 694310 947466 946384 724802 706649 145024 > 2243 [i]