Best Known (52, 115, s)-Nets in Base 2
(52, 115, 36)-Net over F2 — Constructive and digital
Digital (52, 115, 36)-net over F2, using
- t-expansion [i] based on digital (51, 115, 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]
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
(52, 115, 40)-Net over F2 — Digital
Digital (52, 115, 40)-net over F2, using
- t-expansion [i] based on digital (50, 115, 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
(52, 115, 113)-Net over F2 — Upper bound on s (digital)
There is no digital (52, 115, 114)-net over F2, because
- 7 times m-reduction [i] would yield digital (52, 108, 114)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2108, 114, F2, 56) (dual of [114, 6, 57]-code), but
(52, 115, 114)-Net in Base 2 — Upper bound on s
There is no (52, 115, 115)-net in base 2, because
- 5 times m-reduction [i] would yield (52, 110, 115)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2110, 115, S2, 58), but
- adding a parity check bit [i] would yield OA(2111, 116, S2, 59), but
- the (dual) Plotkin bound shows that M ≥ 41538 374868 278621 028243 970633 760768 / 15 > 2111 [i]
- adding a parity check bit [i] would yield OA(2111, 116, S2, 59), but
- extracting embedded orthogonal array [i] would yield OA(2110, 115, S2, 58), but