Best Known (54, 54+67, s)-Nets in Base 2
(54, 54+67, 42)-Net over F2 — Constructive and digital
Digital (54, 121, 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+67, 117)-Net over F2 — Upper bound on s (digital)
There is no digital (54, 121, 118)-net over F2, because
- 11 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+67, 118)-Net in Base 2 — Upper bound on s
There is no (54, 121, 119)-net in base 2, because
- 7 times m-reduction [i] would yield (54, 114, 119)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2114, 119, S2, 60), but
- adding a parity check bit [i] would yield OA(2115, 120, S2, 61), but
- the (dual) Plotkin bound shows that M ≥ 1 329227 995784 915872 903807 060280 344576 / 31 > 2115 [i]
- adding a parity check bit [i] would yield OA(2115, 120, S2, 61), but
- extracting embedded orthogonal array [i] would yield OA(2114, 119, S2, 60), but