MinT - Best Known (29, 122, s)-Nets in Base 5

Best Known (29, 122, s)-Nets in Base 5

(29, 122, 51)-Net over F₅ — Constructive and digital

Digital (29, 122, 51)-net over F₅, using

t-expansion [i] based on digital (22, 122, 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]

(29, 122, 56)-Net over F₅ — Digital

Digital (29, 122, 56)-net over F₅, using

net from sequence [i] based on digital (29, 55)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 29 and N(F) ≥ 56, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(29, 122, 229)-Net over F₅ — Upper bound on s (digital)

There is no digital (29, 122, 230)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹²², 230, F₅, 93) (dual of [230, 108, 94]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹²¹, 148, S₅, 93), but
    - the linear programming bound shows that MÂ â‰¥ 32369 614861 849619 401750 006027 906460 896693 183159 168094 972472 912293 146030 116020 028799 539431 929588 317871 093750 / 5704 050953 652436 468413 > 5¹²¹ [i]
  2. linear OA(5¹⁰⁸, 230, F₅, 82) (dual of [230, 122, 83]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5¹⁰⁸, 228, F₅, 82) (dual of [228, 120, 83]-code), but
      - constructionÂ Y1 [i] would yield
        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]
        OA(5¹²⁰, 228, S₅, 92), but
        discarding factors would yield OA(5¹²⁰, 149, S₅, 92), but
        the linear programming bound shows that MÂ â‰¥ 1875 115433 709844 937609 244603 657370 223629 726878 856922 329955 321895 357533 978909 714278 415776 789188 385009 765625 / 2299 458852 249280 370649 > 5¹²⁰ [i]

(29, 122, 277)-Net in Base 5 — Upper bound on s

There is no (29, 122, 278)-net in base 5, because

1 times m-reduction [i] would yield (29, 121, 278)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4 023653 029074 042490 044441 594173 427807 065601 408794 029906 039425 899070 071537 091422 220225 > 5¹²¹ [i]