Best Known (6−4, 6, s)-Nets in Base 5
(6−4, 6, 27)-Net over F5 — Constructive and digital
Digital (2, 6, 27)-net over F5, using
(6−4, 6, 42)-Net over F5 — Upper bound on s (digital)
There is no digital (2, 6, 43)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(56, 43, F5, 4) (dual of [43, 37, 5]-code), but
- construction Y1 [i] would yield
- linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- linear OA(537, 43, F5, 30) (dual of [43, 6, 31]-code), but
- residual code [i] would yield linear OA(57, 12, F5, 6) (dual of [12, 5, 7]-code), but
- linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- construction Y1 [i] would yield
(6−4, 6, 43)-Net in Base 5 — Upper bound on s
There is no (2, 6, 44)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 16193 > 56 [i]
- extracting embedded orthogonal array [i] would yield OA(56, 44, S5, 4), but
- the linear programming bound shows that M ≥ 439375 / 27 > 56 [i]