Best Known (89−11, 89, s)-Nets in Base 2
(89−11, 89, 26218)-Net over F2 — Constructive and digital
Digital (78, 89, 26218)-net over F2, using
- 22 times duplication [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
(89−11, 89, 32773)-Net over F2 — Digital
Digital (78, 89, 32773)-net over F2, using
- 21 times duplication [i] based on digital (77, 88, 32773)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(288, 32773, F2, 4, 11) (dual of [(32773, 4), 131004, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(288, 131092, F2, 11) (dual of [131092, 131004, 12]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(287, 131091, F2, 11) (dual of [131091, 131004, 12]-code), using
- OOA 4-folding [i] based on linear OA(288, 131092, F2, 11) (dual of [131092, 131004, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(288, 32773, F2, 4, 11) (dual of [(32773, 4), 131004, 12]-NRT-code), using
(89−11, 89, 517557)-Net in Base 2 — Upper bound on s
There is no (78, 89, 517558)-net in base 2, because
- 1 times m-reduction [i] would yield (78, 88, 517558)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 309 487599 616745 831178 691503 > 288 [i]