Best Known (226, 226+33, s)-Nets in Base 2
(226, 226+33, 4097)-Net over F2 — Constructive and digital
Digital (226, 259, 4097)-net over F2, using
- 21 times duplication [i] based on digital (225, 258, 4097)-net over F2, using
- net defined by OOA [i] based on linear OOA(2258, 4097, F2, 33, 33) (dual of [(4097, 33), 134943, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(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(32) ⊂ Ce(30) [i] based on
- OOA 16-folding and stacking with additional row [i] based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- net defined by OOA [i] based on linear OOA(2258, 4097, F2, 33, 33) (dual of [(4097, 33), 134943, 34]-NRT-code), using
(226, 226+33, 9364)-Net over F2 — Digital
Digital (226, 259, 9364)-net over F2, using
- 21 times duplication [i] based on digital (225, 258, 9364)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 9364, F2, 7, 33) (dual of [(9364, 7), 65290, 34]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2258, 65548, F2, 33) (dual of [65548, 65290, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(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(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2258, 65553, F2, 33) (dual of [65553, 65295, 34]-code), using
- OOA 7-folding [i] based on linear OA(2258, 65548, F2, 33) (dual of [65548, 65290, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 9364, F2, 7, 33) (dual of [(9364, 7), 65290, 34]-NRT-code), using
(226, 226+33, 485988)-Net in Base 2 — Upper bound on s
There is no (226, 259, 485989)-net in base 2, because
- 1 times m-reduction [i] would yield (226, 258, 485989)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463168 938661 790566 174457 593828 689494 953633 588400 996173 889757 902489 816840 689050 > 2258 [i]