Best Known (104−11, 104, s)-Nets in Base 2
(104−11, 104, 209719)-Net over F2 — Constructive and digital
Digital (93, 104, 209719)-net over F2, using
- 22 times duplication [i] based on digital (91, 102, 209719)-net over F2, using
- net defined by OOA [i] based on linear OOA(2102, 209719, F2, 11, 11) (dual of [(209719, 11), 2306807, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2102, 1048596, F2, 11) (dual of [1048596, 1048494, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2102, 1048597, F2, 11) (dual of [1048597, 1048495, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(2101, 1048576, F2, 11) (dual of [1048576, 1048475, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(281, 1048576, F2, 9) (dual of [1048576, 1048495, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(2102, 1048597, F2, 11) (dual of [1048597, 1048495, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2102, 1048596, F2, 11) (dual of [1048596, 1048494, 12]-code), using
- net defined by OOA [i] based on linear OOA(2102, 209719, F2, 11, 11) (dual of [(209719, 11), 2306807, 12]-NRT-code), using
(104−11, 104, 262150)-Net over F2 — Digital
Digital (93, 104, 262150)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 262150, F2, 4, 11) (dual of [(262150, 4), 1048496, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2104, 1048600, F2, 11) (dual of [1048600, 1048496, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2102, 1048598, F2, 11) (dual of [1048598, 1048496, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(2101, 1048576, F2, 11) (dual of [1048576, 1048475, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(281, 1048576, F2, 9) (dual of [1048576, 1048495, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2102, 1048598, F2, 11) (dual of [1048598, 1048496, 12]-code), using
- OOA 4-folding [i] based on linear OA(2104, 1048600, F2, 11) (dual of [1048600, 1048496, 12]-code), using
(104−11, 104, 4140506)-Net in Base 2 — Upper bound on s
There is no (93, 104, 4140507)-net in base 2, because
- 1 times m-reduction [i] would yield (93, 103, 4140507)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 10 141216 189990 366995 555587 648768 > 2103 [i]