MinT - Best Known (30, 30+97, s)-Nets in Base 5

Best Known (30, 30+97, s)-Nets in Base 5

(30, 30+97, 51)-Net over F₅ — Constructive and digital

Digital (30, 127, 51)-net over F₅, using

t-expansion [i] based on digital (22, 127, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(30, 30+97, 58)-Net over F₅ — Digital

Digital (30, 127, 58)-net over F₅, using

net from sequence [i] based on digital (30, 57)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 30 and N(F) ≥ 58, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(30, 30+97, 221)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 127, 222)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹²⁷, 222, F₅, 97) (dual of [222, 95, 98]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹²⁶, 150, S₅, 97), but
    - the linear programming bound shows that MÂ â‰¥ 36 026079 456021 401807 410154 480760 438768 372858 795374 723787 651793 517725 247462 113458 141175 215132 534503 936767 578125 / 2872 765687 918138 939304 > 5¹²⁶ [i]
  2. OA(5⁹⁵, 222, S₅, 72), but
    - discarding factors would yield OA(5⁹⁵, 145, S₅, 72), but
      - the linear programming bound shows that MÂ â‰¥ 681 594308 011084 642541 521008 533375 736738 792368 662881 901785 086204 561152 741164 839513 298410 490771 406244 903118 931688 368320 465087 890625 / 251 826768 360176 313961 784872 463983 728554 549094 621216 187309 311959 > 5⁹⁵ [i]

(30, 30+97, 286)-Net in Base 5 — Upper bound on s

There is no (30, 127, 287)-net in base 5, because

1 times m-reduction [i] would yield (30, 126, 287)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 13445 475431 041643 744312 526419 670045 028208 988288 892769 290545 163100 224467 988876 371096 866625 > 5¹²⁶ [i]