Best Known (181, 181+21, s)-Nets in Base 2
(181, 181+21, 104859)-Net over F2 — Constructive and digital
Digital (181, 202, 104859)-net over F2, using
- net defined by OOA [i] based on linear OOA(2202, 104859, F2, 21, 21) (dual of [(104859, 21), 2201837, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2202, 1048591, F2, 21) (dual of [1048591, 1048389, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2202, 1048591, F2, 21) (dual of [1048591, 1048389, 22]-code), using
(181, 181+21, 149799)-Net over F2 — Digital
Digital (181, 202, 149799)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 149799, F2, 7, 21) (dual of [(149799, 7), 1048391, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2202, 1048593, F2, 21) (dual of [1048593, 1048391, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- OOA 7-folding [i] based on linear OA(2202, 1048593, F2, 21) (dual of [1048593, 1048391, 22]-code), using
(181, 181+21, 5089533)-Net in Base 2 — Upper bound on s
There is no (181, 202, 5089534)-net in base 2, because
- 1 times m-reduction [i] would yield (181, 201, 5089534)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 213879 131621 817484 172787 211048 928618 165377 930593 675049 579738 > 2201 [i]