Best Known (241−30, 241, s)-Nets in Base 2
(241−30, 241, 4370)-Net over F2 — Constructive and digital
Digital (211, 241, 4370)-net over F2, using
- net defined by OOA [i] based on linear OOA(2241, 4370, F2, 30, 30) (dual of [(4370, 30), 130859, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2241, 65550, F2, 30) (dual of [65550, 65309, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 65552, F2, 30) (dual of [65552, 65311, 31]-code), using
- 1 times truncation [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 65552, F2, 30) (dual of [65552, 65311, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2241, 65550, F2, 30) (dual of [65550, 65309, 31]-code), using
(241−30, 241, 10925)-Net over F2 — Digital
Digital (211, 241, 10925)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 10925, F2, 6, 30) (dual of [(10925, 6), 65309, 31]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2241, 65550, F2, 30) (dual of [65550, 65309, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 65552, F2, 30) (dual of [65552, 65311, 31]-code), using
- 1 times truncation [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 65552, F2, 30) (dual of [65552, 65311, 31]-code), using
- OOA 6-folding [i] based on linear OA(2241, 65550, F2, 30) (dual of [65550, 65309, 31]-code), using
(241−30, 241, 440852)-Net in Base 2 — Upper bound on s
There is no (211, 241, 440853)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 533722 743271 942740 742369 099392 430575 909597 254770 568859 810384 394483 920368 > 2241 [i]