Best Known (26−25, 26, s)-Nets in Base 27
(26−25, 26, 38)-Net over F27 — Constructive and digital
Digital (1, 26, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
(26−25, 26, 122)-Net over F27 — Upper bound on s (digital)
There is no digital (1, 26, 123)-net over F27, because
- extracting embedded orthogonal array [i] would yield linear OA(2726, 123, F27, 25) (dual of [123, 97, 26]-code), but
- construction Y1 [i] would yield
- linear OA(2725, 29, F27, 25) (dual of [29, 4, 26]-code or 29-arc in PG(24,27)), but
- OA(2797, 123, S27, 94), but
- discarding factors would yield OA(2797, 103, S27, 94), but
- the linear programming bound shows that M ≥ 128 911025 055144 003712 512083 846133 074073 104655 021247 977332 109220 807942 928575 240457 586722 133506 291651 511625 988777 715999 325631 713723 671307 153546 411869 / 17 583797 > 2797 [i]
- discarding factors would yield OA(2797, 103, S27, 94), but
- construction Y1 [i] would yield
(26−25, 26, 135)-Net in Base 27 — Upper bound on s
There is no (1, 26, 136)-net in base 27, because
- 17 times m-reduction [i] would yield (1, 9, 136)-net in base 27, but
- extracting embedded orthogonal array [i] would yield OA(279, 136, S27, 8), but
- the linear programming bound shows that M ≥ 35766 310676 101581 380904 / 4611 122641 > 279 [i]
- extracting embedded orthogonal array [i] would yield OA(279, 136, S27, 8), but