Best Known (2, 63, s)-Nets in Base 27
(2, 63, 48)-Net over F27 — Constructive and digital
Digital (2, 63, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
(2, 63, 83)-Net over F27 — Upper bound on s (digital)
There is no digital (2, 63, 84)-net over F27, because
- 7 times m-reduction [i] would yield digital (2, 56, 84)-net over F27, but
- extracting embedded orthogonal array [i] would yield linear OA(2756, 84, F27, 54) (dual of [84, 28, 55]-code), but
- residual code [i] would yield OA(272, 29, S27, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 755 > 272 [i]
- residual code [i] would yield OA(272, 29, S27, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(2756, 84, F27, 54) (dual of [84, 28, 55]-code), but
(2, 63, 158)-Net in Base 27 — Upper bound on s
There is no (2, 63, 159)-net in base 27, because
- extracting embedded orthogonal array [i] would yield OA(2763, 159, S27, 61), but
- the linear programming bound shows that M ≥ 699739 490040 343712 958447 540771 554852 668452 883476 668505 377718 410484 768728 972948 002286 694405 392173 219635 / 452558 557757 > 2763 [i]