Best Known (33−4, 33, s)-Nets in Base 2
(33−4, 33, 65537)-Net over F2 — Constructive and digital
Digital (29, 33, 65537)-net over F2, using
(33−4, 33, 65553)-Net over F2 — Digital
Digital (29, 33, 65553)-net over F2, using
- net defined by OOA [i] based on linear OOA(233, 65553, F2, 4, 4) (dual of [(65553, 4), 262179, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(233, 65553, F2, 3, 4) (dual of [(65553, 3), 196626, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(233, 65553, F2, 4) (dual of [65553, 65520, 5]-code), using
- 1 times truncation [i] based on linear OA(234, 65554, F2, 5) (dual of [65554, 65520, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(233, 65536, F2, 5) (dual of [65536, 65503, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(217, 65536, F2, 3) (dual of [65536, 65519, 4]-code or 65536-cap in PG(16,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(234, 65554, F2, 5) (dual of [65554, 65520, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(233, 65553, F2, 4) (dual of [65553, 65520, 5]-code), using
- appending kth column [i] based on linear OOA(233, 65553, F2, 3, 4) (dual of [(65553, 3), 196626, 5]-NRT-code), using
(33−4, 33, 131069)-Net in Base 2 — Upper bound on s
There is no (29, 33, 131070)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 8590 000126 > 233 [i]