Best Known (167−19, 167, s)-Nets in Base 2
(167−19, 167, 29129)-Net over F2 — Constructive and digital
Digital (148, 167, 29129)-net over F2, using
- 23 times duplication [i] based on digital (145, 164, 29129)-net over F2, using
- net defined by OOA [i] based on linear OOA(2164, 29129, F2, 19, 19) (dual of [(29129, 19), 553287, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2164, 262162, F2, 19) (dual of [262162, 261998, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2164, 262163, F2, 19) (dual of [262163, 261999, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2145, 262144, F2, 17) (dual of [262144, 261999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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 X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2164, 262163, F2, 19) (dual of [262163, 261999, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2164, 262162, F2, 19) (dual of [262162, 261998, 20]-code), using
- net defined by OOA [i] based on linear OOA(2164, 29129, F2, 19, 19) (dual of [(29129, 19), 553287, 20]-NRT-code), using
(167−19, 167, 43694)-Net over F2 — Digital
Digital (148, 167, 43694)-net over F2, using
- 23 times duplication [i] based on digital (145, 164, 43694)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2164, 43694, F2, 6, 19) (dual of [(43694, 6), 262000, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2164, 262164, F2, 19) (dual of [262164, 262000, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2145, 262144, F2, 17) (dual of [262144, 261999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [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 X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- OOA 6-folding [i] based on linear OA(2164, 262164, F2, 19) (dual of [262164, 262000, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2164, 43694, F2, 6, 19) (dual of [(43694, 6), 262000, 20]-NRT-code), using
(167−19, 167, 1479376)-Net in Base 2 — Upper bound on s
There is no (148, 167, 1479377)-net in base 2, because
- 1 times m-reduction [i] would yield (148, 166, 1479377)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 93 536491 487082 914141 443198 239268 021254 696280 902316 > 2166 [i]