Best Known (82−81, 82, s)-Nets in Base 64
(82−81, 82, 80)-Net over F64 — Constructive and digital
Digital (1, 82, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
(82−81, 82, 81)-Net over F64 — Digital
Digital (1, 82, 81)-net over F64, using
- net from sequence [i] based on digital (1, 80)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 81, using
(82−81, 82, 191)-Net over F64 — Upper bound on s (digital)
There is no digital (1, 82, 192)-net over F64, because
- 17 times m-reduction [i] would yield digital (1, 65, 192)-net over F64, but
- extracting embedded orthogonal array [i] would yield linear OA(6465, 192, F64, 64) (dual of [192, 127, 65]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(64127, 192, F64, 126) (dual of [192, 65, 127]-code), but
- discarding factors / shortening the dual code would yield linear OA(64127, 130, F64, 126) (dual of [130, 3, 127]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(64127, 192, F64, 126) (dual of [192, 65, 127]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(6465, 192, F64, 64) (dual of [192, 127, 65]-code), but
(82−81, 82, 386)-Net in Base 64 — Upper bound on s
There is no (1, 82, 387)-net in base 64, because
- 69 times m-reduction [i] would yield (1, 13, 387)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 303643 134427 673040 438880 > 6413 [i]