Best Known (70, 70+8, s)-Nets in Base 2
(70, 70+8, 131077)-Net over F2 — Constructive and digital
Digital (70, 78, 131077)-net over F2, using
- t-expansion [i] based on digital (69, 78, 131077)-net over F2, using
- net defined by OOA [i] based on linear OOA(278, 131077, F2, 9, 9) (dual of [(131077, 9), 1179615, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(278, 131077, F2, 8, 9) (dual of [(131077, 8), 1048538, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(278, 524309, F2, 9) (dual of [524309, 524231, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(277, 524288, F2, 9) (dual of [524288, 524211, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(258, 524288, F2, 7) (dual of [524288, 524230, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(278, 524309, F2, 9) (dual of [524309, 524231, 10]-code), using
- appending kth column [i] based on linear OOA(278, 131077, F2, 8, 9) (dual of [(131077, 8), 1048538, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(278, 131077, F2, 9, 9) (dual of [(131077, 9), 1179615, 10]-NRT-code), using
(70, 70+8, 174769)-Net over F2 — Digital
Digital (70, 78, 174769)-net over F2, using
- 21 times duplication [i] based on digital (69, 77, 174769)-net over F2, using
- net defined by OOA [i] based on linear OOA(277, 174769, F2, 8, 8) (dual of [(174769, 8), 1398075, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(277, 174769, F2, 7, 8) (dual of [(174769, 7), 1223306, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 174769, F2, 3, 8) (dual of [(174769, 3), 524230, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(277, 524307, F2, 8) (dual of [524307, 524230, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(277, 524308, F2, 8) (dual of [524308, 524231, 9]-code), using
- 1 times truncation [i] based on linear OA(278, 524309, F2, 9) (dual of [524309, 524231, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(277, 524288, F2, 9) (dual of [524288, 524211, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(258, 524288, F2, 7) (dual of [524288, 524230, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(278, 524309, F2, 9) (dual of [524309, 524231, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(277, 524308, F2, 8) (dual of [524308, 524231, 9]-code), using
- OOA 3-folding [i] based on linear OA(277, 524307, F2, 8) (dual of [524307, 524230, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(277, 174769, F2, 3, 8) (dual of [(174769, 3), 524230, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(277, 174769, F2, 7, 8) (dual of [(174769, 7), 1223306, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(277, 174769, F2, 8, 8) (dual of [(174769, 8), 1398075, 9]-NRT-code), using
(70, 70+8, 1641104)-Net in Base 2 — Upper bound on s
There is no (70, 78, 1641105)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 302231 728464 626301 899596 > 278 [i]