Best Known (37, 50, s)-Nets in Base 2
(37, 50, 73)-Net over F2 — Constructive and digital
Digital (37, 50, 73)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (9, 15, 36)-net over F2, using
- digital (22, 35, 37)-net over F2, using
- 2 times m-reduction [i] based on digital (22, 37, 37)-net over F2, using
(37, 50, 113)-Net over F2 — Digital
Digital (37, 50, 113)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(250, 113, F2, 2, 13) (dual of [(113, 2), 176, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(250, 132, F2, 2, 13) (dual of [(132, 2), 214, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(250, 264, F2, 13) (dual of [264, 214, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(250, 265, F2, 13) (dual of [265, 215, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(249, 256, F2, 13) (dual of [256, 207, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(241, 256, F2, 11) (dual of [256, 215, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(250, 265, F2, 13) (dual of [265, 215, 14]-code), using
- OOA 2-folding [i] based on linear OA(250, 264, F2, 13) (dual of [264, 214, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(250, 132, F2, 2, 13) (dual of [(132, 2), 214, 14]-NRT-code), using
(37, 50, 851)-Net in Base 2 — Upper bound on s
There is no (37, 50, 852)-net in base 2, because
- 1 times m-reduction [i] would yield (37, 49, 852)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 563 718810 047516 > 249 [i]