Best Known (178−30, 178, s)-Nets in Base 2
(178−30, 178, 380)-Net over F2 — Constructive and digital
Digital (148, 178, 380)-net over F2, using
- 23 times duplication [i] based on digital (145, 175, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 35, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 35, 76)-net over F32, using
(178−30, 178, 963)-Net over F2 — Digital
Digital (148, 178, 963)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 963, F2, 2, 30) (dual of [(963, 2), 1748, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 1046, F2, 2, 30) (dual of [(1046, 2), 1914, 31]-NRT-code), using
- strength reduction [i] based on linear OOA(2178, 1046, F2, 2, 31) (dual of [(1046, 2), 1914, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2178, 2092, F2, 31) (dual of [2092, 1914, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2178, 2093, F2, 31) (dual of [2093, 1915, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2178, 2093, F2, 31) (dual of [2093, 1915, 32]-code), using
- OOA 2-folding [i] based on linear OA(2178, 2092, F2, 31) (dual of [2092, 1914, 32]-code), using
- strength reduction [i] based on linear OOA(2178, 1046, F2, 2, 31) (dual of [(1046, 2), 1914, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 1046, F2, 2, 30) (dual of [(1046, 2), 1914, 31]-NRT-code), using
(178−30, 178, 23965)-Net in Base 2 — Upper bound on s
There is no (148, 178, 23966)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 383185 008409 978887 368443 587375 562010 754047 566345 933068 > 2178 [i]