Best Known (212, 212+31, s)-Nets in Base 2
(212, 212+31, 4370)-Net over F2 — Constructive and digital
Digital (212, 243, 4370)-net over F2, using
- 21 times duplication [i] based on digital (211, 242, 4370)-net over F2, using
- net defined by OOA [i] based on linear OOA(2242, 4370, F2, 31, 31) (dual of [(4370, 31), 135228, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2242, 65551, F2, 31) (dual of [65551, 65309, 32]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2242, 65551, F2, 31) (dual of [65551, 65309, 32]-code), using
- net defined by OOA [i] based on linear OOA(2242, 4370, F2, 31, 31) (dual of [(4370, 31), 135228, 32]-NRT-code), using
(212, 212+31, 9364)-Net over F2 — Digital
Digital (212, 243, 9364)-net over F2, using
- 21 times duplication [i] based on digital (211, 242, 9364)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 9364, F2, 7, 31) (dual of [(9364, 7), 65306, 32]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2242, 65548, F2, 31) (dual of [65548, 65306, 32]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- OOA 7-folding [i] based on linear OA(2242, 65548, F2, 31) (dual of [65548, 65306, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 9364, F2, 7, 31) (dual of [(9364, 7), 65306, 32]-NRT-code), using
(212, 212+31, 461703)-Net in Base 2 — Upper bound on s
There is no (212, 243, 461704)-net in base 2, because
- 1 times m-reduction [i] would yield (212, 242, 461704)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 7 067492 243895 399586 990088 126705 609083 252603 234670 536124 645172 175477 386448 > 2242 [i]