Best Known (126−96, 126, s)-Nets in Base 5
(126−96, 126, 51)-Net over F5 — Constructive and digital
Digital (30, 126, 51)-net over F5, using
- t-expansion [i] based on digital (22, 126, 51)-net over F5, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 22 and N(F) ≥ 51, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
(126−96, 126, 58)-Net over F5 — Digital
Digital (30, 126, 58)-net over F5, using
- net from sequence [i] based on digital (30, 57)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 30 and N(F) ≥ 58, using
(126−96, 126, 284)-Net over F5 — Upper bound on s (digital)
There is no digital (30, 126, 285)-net over F5, because
- 1 times m-reduction [i] would yield digital (30, 125, 285)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(5125, 285, F5, 95) (dual of [285, 160, 96]-code), but
- residual code [i] would yield OA(530, 189, S5, 19), but
- 1 times truncation [i] would yield OA(529, 188, S5, 18), but
- the linear programming bound shows that M ≥ 22 829644 377403 021434 618461 608886 718750 / 121561 995449 060039 > 529 [i]
- 1 times truncation [i] would yield OA(529, 188, S5, 18), but
- residual code [i] would yield OA(530, 189, S5, 19), but
- extracting embedded orthogonal array [i] would yield linear OA(5125, 285, F5, 95) (dual of [285, 160, 96]-code), but
(126−96, 126, 286)-Net in Base 5 — Upper bound on s
There is no (30, 126, 287)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 13445 475431 041643 744312 526419 670045 028208 988288 892769 290545 163100 224467 988876 371096 866625 > 5126 [i]