Best Known (83, 83+84, s)-Nets in Base 2
(83, 83+84, 51)-Net over F2 — Constructive and digital
Digital (83, 167, 51)-net over F2, using
- t-expansion [i] based on digital (80, 167, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(83, 83+84, 57)-Net over F2 — Digital
Digital (83, 167, 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
(83, 83+84, 177)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 167, 178)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2167, 178, F2, 84) (dual of [178, 11, 85]-code), but
- residual code [i] would yield linear OA(283, 93, F2, 42) (dual of [93, 10, 43]-code), but
- residual code [i] would yield linear OA(241, 50, F2, 21) (dual of [50, 9, 22]-code), but
- 1 times truncation [i] would yield linear OA(240, 49, F2, 20) (dual of [49, 9, 21]-code), but
- residual code [i] would yield linear OA(241, 50, F2, 21) (dual of [50, 9, 22]-code), but
- residual code [i] would yield linear OA(283, 93, F2, 42) (dual of [93, 10, 43]-code), but
(83, 83+84, 203)-Net in Base 2 — Upper bound on s
There is no (83, 167, 204)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 204 727916 359907 742101 899783 607952 602857 929818 334750 > 2167 [i]