Best Known (151, 151+21, s)-Nets in Base 2
(151, 151+21, 13108)-Net over F2 — Constructive and digital
Digital (151, 172, 13108)-net over F2, using
- net defined by OOA [i] based on linear OOA(2172, 13108, F2, 21, 21) (dual of [(13108, 21), 275096, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2172, 131081, F2, 21) (dual of [131081, 130909, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2172, 131081, F2, 21) (dual of [131081, 130909, 22]-code), using
(151, 151+21, 21848)-Net over F2 — Digital
Digital (151, 172, 21848)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 21848, F2, 6, 21) (dual of [(21848, 6), 130916, 22]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2172, 131088, F2, 21) (dual of [131088, 130916, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- OOA 6-folding [i] based on linear OA(2172, 131088, F2, 21) (dual of [131088, 130916, 22]-code), using
(151, 151+21, 636179)-Net in Base 2 — Upper bound on s
There is no (151, 172, 636180)-net in base 2, because
- 1 times m-reduction [i] would yield (151, 171, 636180)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2993 202288 227603 470870 591746 215824 974550 248887 100314 > 2171 [i]