Best Known (59, 59+63, s)-Nets in Base 2
(59, 59+63, 43)-Net over F2 — Constructive and digital
Digital (59, 122, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
(59, 59+63, 128)-Net over F2 — Upper bound on s (digital)
There is no digital (59, 122, 129)-net over F2, because
- 3 times m-reduction [i] would yield digital (59, 119, 129)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2119, 129, F2, 60) (dual of [129, 10, 61]-code), but
- residual code [i] would yield linear OA(259, 68, F2, 30) (dual of [68, 9, 31]-code), but
- adding a parity check bit [i] would yield linear OA(260, 69, F2, 31) (dual of [69, 9, 32]-code), but
- “BGV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(260, 69, F2, 31) (dual of [69, 9, 32]-code), but
- residual code [i] would yield linear OA(259, 68, F2, 30) (dual of [68, 9, 31]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2119, 129, F2, 60) (dual of [129, 10, 61]-code), but
(59, 59+63, 129)-Net in Base 2 — Upper bound on s
There is no (59, 122, 130)-net in base 2, because
- 3 times m-reduction [i] would yield (59, 119, 130)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2119, 130, S2, 60), but
- the linear programming bound shows that M ≥ 16610 033035 328308 747805 973025 263185 821696 / 21793 > 2119 [i]
- extracting embedded orthogonal array [i] would yield OA(2119, 130, S2, 60), but