Best Known (51, 51+44, s)-Nets in Base 2
(51, 51+44, 36)-Net over F2 — Constructive and digital
Digital (51, 95, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(51, 51+44, 40)-Net over F2 — Digital
Digital (51, 95, 40)-net over F2, using
- t-expansion [i] based on digital (50, 95, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(51, 51+44, 128)-Net over F2 — Upper bound on s (digital)
There is no digital (51, 95, 129)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(295, 129, F2, 44) (dual of [129, 34, 45]-code), but
- adding a parity check bit [i] would yield linear OA(296, 130, F2, 45) (dual of [130, 34, 46]-code), but
(51, 51+44, 129)-Net in Base 2 — Upper bound on s
There is no (51, 95, 130)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(295, 130, S2, 44), but
- the linear programming bound shows that M ≥ 184927 376142 602562 778376 736697 353811 001344 / 3 793040 420631 > 295 [i]