Best Known (31−28, 31, s)-Nets in Base 7
(31−28, 31, 11)-Net over F7 — Constructive and digital
Digital (3, 31, 11)-net over F7, using
- net from sequence [i] based on digital (3, 10)-sequence over F7, using
(31−28, 31, 20)-Net over F7 — Digital
Digital (3, 31, 20)-net over F7, using
- net from sequence [i] based on digital (3, 19)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 3 and N(F) ≥ 20, using
(31−28, 31, 28)-Net over F7 — Upper bound on s (digital)
There is no digital (3, 31, 29)-net over F7, because
- 7 times m-reduction [i] would yield digital (3, 24, 29)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(724, 29, F7, 21) (dual of [29, 5, 22]-code), but
- construction Y1 [i] would yield
- OA(723, 25, S7, 21), but
- the (dual) Plotkin bound shows that M ≥ 383 162462 761132 828802 / 11 > 723 [i]
- OA(75, 29, S7, 4), but
- the linear programming bound shows that M ≥ 3 384381 / 197 > 75 [i]
- OA(723, 25, S7, 21), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(724, 29, F7, 21) (dual of [29, 5, 22]-code), but
(31−28, 31, 32)-Net in Base 7 — Upper bound on s
There is no (3, 31, 33)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(731, 33, S7, 28), but
- the (dual) Plotkin bound shows that M ≥ 5522 138371 219603 231526 496005 / 29 > 731 [i]