Best Known (53, 110, s)-Nets in Base 2
(53, 110, 36)-Net over F2 — Constructive and digital
Digital (53, 110, 36)-net over F2, using
- t-expansion [i] based on digital (51, 110, 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
(53, 110, 40)-Net over F2 — Digital
Digital (53, 110, 40)-net over F2, using
- t-expansion [i] based on digital (50, 110, 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
(53, 110, 116)-Net over F2 — Upper bound on s (digital)
There is no digital (53, 110, 117)-net over F2, because
- 1 times m-reduction [i] would yield digital (53, 109, 117)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2109, 117, F2, 56) (dual of [117, 8, 57]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2110, 118, F2, 56) (dual of [118, 8, 57]-code), but
- adding a parity check bit [i] would yield linear OA(2111, 119, F2, 57) (dual of [119, 8, 58]-code), but
- “DMa†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(2111, 119, F2, 57) (dual of [119, 8, 58]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2110, 118, F2, 56) (dual of [118, 8, 57]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2109, 117, F2, 56) (dual of [117, 8, 57]-code), but
(53, 110, 117)-Net in Base 2 — Upper bound on s
There is no (53, 110, 118)-net in base 2, because
- 3 times m-reduction [i] would yield (53, 107, 118)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2107, 118, S2, 54), but
- the linear programming bound shows that M ≥ 6490 371073 168534 535663 120411 525120 / 33 > 2107 [i]
- extracting embedded orthogonal array [i] would yield OA(2107, 118, S2, 54), but