Best Known (88, 99, s)-Nets in Base 2
(88, 99, 104862)-Net over F2 — Constructive and digital
Digital (88, 99, 104862)-net over F2, using
- net defined by OOA [i] based on linear OOA(299, 104862, F2, 11, 11) (dual of [(104862, 11), 1153383, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(299, 524311, F2, 11) (dual of [524311, 524212, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(8) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(299, 524311, F2, 11) (dual of [524311, 524212, 12]-code), using
(88, 99, 131077)-Net over F2 — Digital
Digital (88, 99, 131077)-net over F2, using
- 22 times duplication [i] based on digital (86, 97, 131077)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(297, 131077, F2, 4, 11) (dual of [(131077, 4), 524211, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(297, 524308, F2, 11) (dual of [524308, 524211, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(21, 20, F2, 1) (dual of [20, 19, 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
- OOA 4-folding [i] based on linear OA(297, 524308, F2, 11) (dual of [524308, 524211, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(297, 131077, F2, 4, 11) (dual of [(131077, 4), 524211, 12]-NRT-code), using
(88, 99, 2070249)-Net in Base 2 — Upper bound on s
There is no (88, 99, 2070250)-net in base 2, because
- 1 times m-reduction [i] would yield (88, 98, 2070250)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 316913 005931 376040 848246 373426 > 298 [i]