Best Known (31, 31+33, s)-Nets in Base 81
(31, 31+33, 370)-Net over F81 — Constructive and digital
Digital (31, 64, 370)-net over F81, using
- t-expansion [i] based on digital (16, 64, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(31, 31+33, 1159)-Net over F81 — Digital
Digital (31, 64, 1159)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8164, 1159, F81, 33) (dual of [1159, 1095, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 1313, F81, 33) (dual of [1313, 1249, 34]-code), using
(31, 31+33, 2780400)-Net in Base 81 — Upper bound on s
There is no (31, 64, 2780401)-net in base 81, because
- 1 times m-reduction [i] would yield (31, 63, 2780401)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 1 716162 540904 510763 719618 353961 258422 047458 505737 933898 051492 387981 106796 500341 716142 766644 523497 677896 513858 979948 801281 > 8163 [i]