MinT - Best Known (29, 124, s)-Nets in Base 5

Best Known (29, 124, s)-Nets in Base 5

(29, 124, 51)-Net over F₅ — Constructive and digital

Digital (29, 124, 51)-net over F₅, using

t-expansion [i] based on digital (22, 124, 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, 124, 56)-Net over F₅ — Digital

Digital (29, 124, 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, 124, 209)-Net over F₅ — Upper bound on s (digital)

There is no digital (29, 124, 210)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹²⁴, 210, F₅, 95) (dual of [210, 86, 96]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹²³, 145, S₅, 95), but
    - the linear programming bound shows that MÂ â‰¥ 523946 922524 269524 545786 907827 294722 187715 168920 019132 326750 498085 399243 564097 560010 850429 534912 109375 / 4025 476446 595648 > 5¹²³ [i]
  2. OA(5⁸⁶, 210, S₅, 65), but
    - discarding factors would yield OA(5⁸⁶, 145, S₅, 65), but
      - the linear programming bound shows that MÂ â‰¥ 1 681971 586304 181223 187907 190141 264188 172812 749344 276629 843014 476462 881651 126808 966524 933987 666420 440086 867329 609762 499671 015636 681116 117777 216867 334999 506056 192331 016063 690185 546875 / 1 235197 506395 108210 300700 030853 409382 253498 637886 138574 180390 146599 149915 868081 543662 842688 306834 854616 250416 282850 014131 > 5⁸⁶ [i]

(29, 124, 276)-Net in Base 5 — Upper bound on s

There is no (29, 124, 277)-net in base 5, because

1 times m-reduction [i] would yield (29, 123, 277)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 101 602272 954304 723630 806409 035219 158151 467971 289945 473724 680826 015922 071314 676159 284125 > 5¹²³ [i]