Best Known (146, 146+16, s)-Nets in Base 2
(146, 146+16, 131074)-Net over F2 — Constructive and digital
Digital (146, 162, 131074)-net over F2, using
- t-expansion [i] based on digital (145, 162, 131074)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 131074, F2, 17, 17) (dual of [(131074, 17), 2228096, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2162, 1048593, F2, 17) (dual of [1048593, 1048431, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2162, 1048593, F2, 17) (dual of [1048593, 1048431, 18]-code), using
- net defined by OOA [i] based on linear OOA(2162, 131074, F2, 17, 17) (dual of [(131074, 17), 2228096, 18]-NRT-code), using
(146, 146+16, 209719)-Net over F2 — Digital
Digital (146, 162, 209719)-net over F2, using
- 21 times duplication [i] based on digital (145, 161, 209719)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 209719, F2, 5, 16) (dual of [(209719, 5), 1048434, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2161, 1048595, F2, 16) (dual of [1048595, 1048434, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 1048597, F2, 16) (dual of [1048597, 1048436, 17]-code), using
- 1 times truncation [i] based on linear OA(2162, 1048598, F2, 17) (dual of [1048598, 1048436, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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 X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2162, 1048598, F2, 17) (dual of [1048598, 1048436, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 1048597, F2, 16) (dual of [1048597, 1048436, 17]-code), using
- OOA 5-folding [i] based on linear OA(2161, 1048595, F2, 16) (dual of [1048595, 1048434, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 209719, F2, 5, 16) (dual of [(209719, 5), 1048434, 17]-NRT-code), using
(146, 146+16, 4694035)-Net in Base 2 — Upper bound on s
There is no (146, 162, 4694036)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5 846006 804603 863632 326045 975641 340968 330671 465022 > 2162 [i]