Best Known (77, 77+10, s)-Nets in Base 2
(77, 77+10, 26218)-Net over F2 — Constructive and digital
Digital (77, 87, 26218)-net over F2, using
- t-expansion [i] based on digital (76, 87, 26218)-net over F2, using
- net defined by OOA [i] based on linear OOA(287, 26218, F2, 11, 11) (dual of [(26218, 11), 288311, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(287, 131091, F2, 11) (dual of [131091, 131004, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(269, 131072, F2, 9) (dual of [131072, 131003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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
- OOA 5-folding and stacking with additional row [i] based on linear OA(287, 131091, F2, 11) (dual of [131091, 131004, 12]-code), using
- net defined by OOA [i] based on linear OOA(287, 26218, F2, 11, 11) (dual of [(26218, 11), 288311, 12]-NRT-code), using
(77, 77+10, 43697)-Net over F2 — Digital
Digital (77, 87, 43697)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(287, 43697, F2, 3, 10) (dual of [(43697, 3), 131004, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(287, 131091, F2, 10) (dual of [131091, 131004, 11]-code), using
- strength reduction [i] based on linear OA(287, 131091, F2, 11) (dual of [131091, 131004, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(269, 131072, F2, 9) (dual of [131072, 131003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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
- strength reduction [i] based on linear OA(287, 131091, F2, 11) (dual of [131091, 131004, 12]-code), using
- OOA 3-folding [i] based on linear OA(287, 131091, F2, 10) (dual of [131091, 131004, 11]-code), using
(77, 77+10, 450558)-Net in Base 2 — Upper bound on s
There is no (77, 87, 450559)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 154 742937 247564 893012 099068 > 287 [i]