Best Known (90−8, 90, s)-Nets in Base 2
(90−8, 90, 1048582)-Net over F2 — Constructive and digital
Digital (82, 90, 1048582)-net over F2, using
- net defined by OOA [i] based on linear OOA(290, 1048582, F2, 8, 8) (dual of [(1048582, 8), 8388566, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(290, 4194328, F2, 8) (dual of [4194328, 4194238, 9]-code), using
- strength reduction [i] based on linear OA(290, 4194328, F2, 9) (dual of [4194328, 4194238, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(289, 4194304, F2, 9) (dual of [4194304, 4194215, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(267, 4194304, F2, 7) (dual of [4194304, 4194237, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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(8) ⊂ Ce(6) [i] based on
- strength reduction [i] based on linear OA(290, 4194328, F2, 9) (dual of [4194328, 4194238, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(290, 4194328, F2, 8) (dual of [4194328, 4194238, 9]-code), using
(90−8, 90, 1398109)-Net over F2 — Digital
Digital (82, 90, 1398109)-net over F2, using
- 21 times duplication [i] based on digital (81, 89, 1398109)-net over F2, using
- net defined by OOA [i] based on linear OOA(289, 1398109, F2, 8, 8) (dual of [(1398109, 8), 11184783, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(289, 1398109, F2, 7, 8) (dual of [(1398109, 7), 9786674, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(289, 1398109, F2, 3, 8) (dual of [(1398109, 3), 4194238, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(289, 4194327, F2, 8) (dual of [4194327, 4194238, 9]-code), using
- 1 times truncation [i] based on linear OA(290, 4194328, F2, 9) (dual of [4194328, 4194238, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(289, 4194304, F2, 9) (dual of [4194304, 4194215, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(267, 4194304, F2, 7) (dual of [4194304, 4194237, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(290, 4194328, F2, 9) (dual of [4194328, 4194238, 10]-code), using
- OOA 3-folding [i] based on linear OA(289, 4194327, F2, 8) (dual of [4194327, 4194238, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(289, 1398109, F2, 3, 8) (dual of [(1398109, 3), 4194238, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(289, 1398109, F2, 7, 8) (dual of [(1398109, 7), 9786674, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(289, 1398109, F2, 8, 8) (dual of [(1398109, 8), 11184783, 9]-NRT-code), using
(90−8, 90, large)-Net in Base 2 — Upper bound on s
There is no (82, 90, large)-net in base 2, because
- 6 times m-reduction [i] would yield (82, 84, large)-net in base 2, but