Best Known (193−96, 193, s)-Nets in Base 2
(193−96, 193, 54)-Net over F2 — Constructive and digital
Digital (97, 193, 54)-net over F2, using
- t-expansion [i] based on digital (95, 193, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 5 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
(193−96, 193, 65)-Net over F2 — Digital
Digital (97, 193, 65)-net over F2, using
- t-expansion [i] based on digital (95, 193, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(193−96, 193, 205)-Net over F2 — Upper bound on s (digital)
There is no digital (97, 193, 206)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2193, 206, F2, 96) (dual of [206, 13, 97]-code), but
- construction Y1 [i] would yield
- linear OA(2192, 202, F2, 96) (dual of [202, 10, 97]-code), but
- residual code [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(249, 57, F2, 25) (dual of [57, 8, 26]-code), but
- residual code [i] would yield linear OA(248, 56, F2, 24) (dual of [56, 8, 25]-code), but
- residual code [i] would yield linear OA(296, 105, F2, 48) (dual of [105, 9, 49]-code), but
- OA(213, 206, S2, 4), but
- discarding factors would yield OA(213, 128, S2, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 8257 > 213 [i]
- discarding factors would yield OA(213, 128, S2, 4), but
- linear OA(2192, 202, F2, 96) (dual of [202, 10, 97]-code), but
- construction Y1 [i] would yield
(193−96, 193, 239)-Net in Base 2 — Upper bound on s
There is no (97, 193, 240)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 13525 604715 067553 575998 934184 060305 223709 460716 022990 442124 > 2193 [i]