Best Known (6, 31, s)-Nets in Base 5
(6, 31, 21)-Net over F5 — Constructive and digital
Digital (6, 31, 21)-net over F5, using
- net from sequence [i] based on digital (6, 20)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 6 and N(F) ≥ 21, using
(6, 31, 38)-Net over F5 — Upper bound on s (digital)
There is no digital (6, 31, 39)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(531, 39, F5, 25) (dual of [39, 8, 26]-code), but
- construction Y1 [i] would yield
- linear OA(530, 33, F5, 25) (dual of [33, 3, 26]-code), but
- OA(58, 39, S5, 6), but
- discarding factors would yield OA(58, 34, S5, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 392089 > 58 [i]
- discarding factors would yield OA(58, 34, S5, 6), but
- construction Y1 [i] would yield
(6, 31, 53)-Net in Base 5 — Upper bound on s
There is no (6, 31, 54)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(531, 54, S5, 25), but
- the linear programming bound shows that M ≥ 533 144408 114014 077000 319957 733154 296875 / 112351 546294 888251 > 531 [i]