Best Known (144−14, 144, s)-Nets in Base 2
(144−14, 144, 149800)-Net over F2 — Constructive and digital
Digital (130, 144, 149800)-net over F2, using
- net defined by OOA [i] based on linear OOA(2144, 149800, F2, 14, 14) (dual of [(149800, 14), 2097056, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2144, 1048600, F2, 14) (dual of [1048600, 1048456, 15]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2141, 1048597, F2, 14) (dual of [1048597, 1048456, 15]-code), using
- 1 times truncation [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2141, 1048597, F2, 14) (dual of [1048597, 1048456, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2144, 1048600, F2, 14) (dual of [1048600, 1048456, 15]-code), using
(144−14, 144, 209720)-Net over F2 — Digital
Digital (130, 144, 209720)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2144, 209720, F2, 5, 14) (dual of [(209720, 5), 1048456, 15]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2144, 1048600, F2, 14) (dual of [1048600, 1048456, 15]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2141, 1048597, F2, 14) (dual of [1048597, 1048456, 15]-code), using
- 1 times truncation [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2141, 1048597, F2, 14) (dual of [1048597, 1048456, 15]-code), using
- OOA 5-folding [i] based on linear OA(2144, 1048600, F2, 14) (dual of [1048600, 1048456, 15]-code), using
(144−14, 144, 5266655)-Net in Base 2 — Upper bound on s
There is no (130, 144, 5266656)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 22 300772 350127 040061 346444 203289 279158 446217 > 2144 [i]