Best Known (51, 51+58, s)-Nets in Base 2
(51, 51+58, 36)-Net over F2 — Constructive and digital
Digital (51, 109, 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]
(51, 51+58, 40)-Net over F2 — Digital
Digital (51, 109, 40)-net over F2, using
- t-expansion [i] based on digital (50, 109, 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
(51, 51+58, 111)-Net over F2 — Upper bound on s (digital)
There is no digital (51, 109, 112)-net over F2, because
- 6 times m-reduction [i] would yield digital (51, 103, 112)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2103, 112, F2, 52) (dual of [112, 9, 53]-code), but
- residual code [i] would yield linear OA(251, 59, F2, 26) (dual of [59, 8, 27]-code), but
- adding a parity check bit [i] would yield linear OA(252, 60, F2, 27) (dual of [60, 8, 28]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(252, 60, F2, 27) (dual of [60, 8, 28]-code), but
- residual code [i] would yield linear OA(251, 59, F2, 26) (dual of [59, 8, 27]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2103, 112, F2, 52) (dual of [112, 9, 53]-code), but
(51, 51+58, 112)-Net in Base 2 — Upper bound on s
There is no (51, 109, 113)-net in base 2, because
- 2 times m-reduction [i] would yield (51, 107, 113)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2107, 113, S2, 56), but
- adding a parity check bit [i] would yield OA(2108, 114, S2, 57), but
- the (dual) Plotkin bound shows that M ≥ 10384 593717 069655 257060 992658 440192 / 29 > 2108 [i]
- adding a parity check bit [i] would yield OA(2108, 114, S2, 57), but
- extracting embedded orthogonal array [i] would yield OA(2107, 113, S2, 56), but