Best Known (138−124, 138, s)-Nets in Base 5
(138−124, 138, 35)-Net over F5 — Constructive and digital
Digital (14, 138, 35)-net over F5, using
- net from sequence [i] based on digital (14, 34)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 11, N(F) = 32, and 3 places with degree 2 [i] based on function field F/F5 with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(138−124, 138, 39)-Net over F5 — Digital
Digital (14, 138, 39)-net over F5, using
- net from sequence [i] based on digital (14, 38)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 14 and N(F) ≥ 39, using
(138−124, 138, 77)-Net over F5 — Upper bound on s (digital)
There is no digital (14, 138, 78)-net over F5, because
- 64 times m-reduction [i] would yield digital (14, 74, 78)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(574, 78, F5, 60) (dual of [78, 4, 61]-code), but
- residual code [i] would yield linear OA(514, 17, F5, 12) (dual of [17, 3, 13]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(574, 78, F5, 60) (dual of [78, 4, 61]-code), but
(138−124, 138, 78)-Net in Base 5 — Upper bound on s
There is no (14, 138, 79)-net in base 5, because
- 68 times m-reduction [i] would yield (14, 70, 79)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(570, 79, S5, 56), but
- the linear programming bound shows that M ≥ 5 918642 718939 423619 239903 473498 998209 834098 815917 968750 / 578151 > 570 [i]
- extracting embedded orthogonal array [i] would yield OA(570, 79, S5, 56), but