Best Known (54, 54+69, s)-Nets in Base 2
(54, 54+69, 42)-Net over F2 — Constructive and digital
Digital (54, 123, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
(54, 54+69, 117)-Net over F2 — Upper bound on s (digital)
There is no digital (54, 123, 118)-net over F2, because
- 13 times m-reduction [i] would yield digital (54, 110, 118)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(2110, 118, F2, 56) (dual of [118, 8, 57]-code), but
(54, 54+69, 118)-Net in Base 2 — Upper bound on s
There is no (54, 123, 119)-net in base 2, because
- 1 times m-reduction [i] would yield (54, 122, 119)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 999816 688399 677449 549061 595709 202115 > 2122 [i]