Best Known (167−29, 167, s)-Nets in Base 2
(167−29, 167, 320)-Net over F2 — Constructive and digital
Digital (138, 167, 320)-net over F2, using
- 22 times duplication [i] based on digital (136, 165, 320)-net over F2, using
- t-expansion [i] based on digital (135, 165, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 33, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 33, 64)-net over F32, using
- t-expansion [i] based on digital (135, 165, 320)-net over F2, using
(167−29, 167, 821)-Net over F2 — Digital
Digital (138, 167, 821)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2167, 821, F2, 2, 29) (dual of [(821, 2), 1475, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 1046, F2, 2, 29) (dual of [(1046, 2), 1925, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2167, 2092, F2, 29) (dual of [2092, 1925, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2167, 2093, F2, 29) (dual of [2093, 1926, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2167, 2093, F2, 29) (dual of [2093, 1926, 30]-code), using
- OOA 2-folding [i] based on linear OA(2167, 2092, F2, 29) (dual of [2092, 1925, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 1046, F2, 2, 29) (dual of [(1046, 2), 1925, 30]-NRT-code), using
(167−29, 167, 22408)-Net in Base 2 — Upper bound on s
There is no (138, 167, 22409)-net in base 2, because
- 1 times m-reduction [i] would yield (138, 166, 22409)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 93 554276 486920 149259 353649 512119 415208 610717 937536 > 2166 [i]