Best Known (89, 89+85, s)-Nets in Base 2
(89, 89+85, 52)-Net over F2 — Constructive and digital
Digital (89, 174, 52)-net over F2, using
- t-expansion [i] based on digital (85, 174, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (85, 51)-sequence over F2, using
(89, 89+85, 57)-Net over F2 — Digital
Digital (89, 174, 57)-net over F2, using
- t-expansion [i] based on digital (83, 174, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(89, 89+85, 194)-Net over F2 — Upper bound on s (digital)
There is no digital (89, 174, 195)-net over F2, because
- 3 times m-reduction [i] would yield digital (89, 171, 195)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2171, 195, F2, 82) (dual of [195, 24, 83]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
- adding a parity check bit [i] would yield linear OA(2173, 197, F2, 83) (dual of [197, 24, 84]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2172, 196, F2, 82) (dual of [196, 24, 83]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2171, 195, F2, 82) (dual of [195, 24, 83]-code), but
(89, 89+85, 229)-Net in Base 2 — Upper bound on s
There is no (89, 174, 230)-net in base 2, because
- 1 times m-reduction [i] would yield (89, 173, 230)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11994 827205 354726 669013 449079 268529 359098 431510 593328 > 2173 [i]