Best Known (30, 30+110, s)-Nets in Base 5
(30, 30+110, 51)-Net over F5 — Constructive and digital
Digital (30, 140, 51)-net over F5, using
- t-expansion [i] based on digital (22, 140, 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
(30, 30+110, 58)-Net over F5 — Digital
Digital (30, 140, 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
(30, 30+110, 163)-Net over F5 — Upper bound on s (digital)
There is no digital (30, 140, 164)-net over F5, because
- 1 times m-reduction [i] would yield digital (30, 139, 164)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(5139, 164, F5, 109) (dual of [164, 25, 110]-code), but
- construction Y1 [i] would yield
- OA(5138, 148, S5, 109), but
- the linear programming bound shows that M ≥ 86 158555 819318 512642 331328 682663 835348 540649 443889 196923 328992 021307 026760 723601 910285 651683 807373 046875 / 29 425011 > 5138 [i]
- OA(525, 164, S5, 16), but
- discarding factors would yield OA(525, 148, S5, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 313081 833470 800945 > 525 [i]
- discarding factors would yield OA(525, 148, S5, 16), but
- OA(5138, 148, S5, 109), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(5139, 164, F5, 109) (dual of [164, 25, 110]-code), but
(30, 30+110, 281)-Net in Base 5 — Upper bound on s
There is no (30, 140, 282)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 73 924718 892573 637638 632005 920278 333822 020975 916813 158186 362674 265482 441031 733496 271321 817794 071945 > 5140 [i]