MinT - Best Known (26, 26+82, s)-Nets in Base 5

Best Known (26, 26+82, s)-Nets in Base 5

(26, 26+82, 51)-Net over F₅ — Constructive and digital

Digital (26, 108, 51)-net over F₅, using

t-expansion [i] based on digital (22, 108, 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]

(26, 26+82, 55)-Net over F₅ — Digital

Digital (26, 108, 55)-net over F₅, using

t-expansion [i] based on digital (23, 108, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(26, 26+82, 226)-Net over F₅ — Upper bound on s (digital)

There is no digital (26, 108, 227)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹⁰⁸, 227, F₅, 82) (dual of [227, 119, 83]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹⁰⁷, 136, S₅, 82), but
    - the linear programming bound shows that MÂ â‰¥ 4 680614 732900 044285 630127 740921 339805 504137 246651 693367 478774 820966 691549 983806 908130 645751 953125 / 7377 156347 408094 956731 > 5¹⁰⁷ [i]
  2. linear OA(5¹¹⁹, 227, F₅, 91) (dual of [227, 108, 92]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5¹¹⁹, 214, F₅, 91) (dual of [214, 95, 92]-code), but
      - constructionÂ Y1 [i] would yield
        OA(5¹¹⁸, 142, S₅, 91), but
        the linear programming bound shows that MÂ â‰¥ 227 290310 028755 622094 163273 225192 363202 339181 888847 417767 861303 072862 710905 610583 722591 400146 484375 / 5712 266942 976429 > 5¹¹⁸ [i]
        OA(5⁹⁵, 214, 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]

(26, 26+82, 250)-Net in Base 5 — Upper bound on s

There is no (26, 108, 251)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3181 444264 236786 306801 714002 978184 882401 905551 905585 646761 148363 188333 346125 > 5¹⁰⁸ [i]