MinT - Best Known (30, 129, s)-Nets in Base 5

Best Known (30, 129, s)-Nets in Base 5

(30, 129, 51)-Net over F₅ — Constructive and digital

Digital (30, 129, 51)-net over F₅, using

t-expansion [i] based on digital (22, 129, 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, 129, 58)-Net over F₅ — Digital

Digital (30, 129, 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, 129, 213)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 129, 214)-net over F₅, because

1 times m-reduction [i] would yield digital (30, 128, 214)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹²⁸, 214, F₅, 98) (dual of [214, 86, 99]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(5¹²⁷, 149, S₅, 98), but
      - the linear programming bound shows that MÂ â‰¥ 5193 361256 119414 558729 044685 549653 156610 735015 094949 701472 602202 230455 237241 812938 009388 744831 085205 078125 / 69869 528585 224173 > 5¹²⁷ [i]
    2. OA(5⁸⁶, 214, 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]

(30, 129, 285)-Net in Base 5 — Upper bound on s

There is no (30, 129, 286)-net in base 5, because

1 times m-reduction [i] would yield (30, 128, 286)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 336430 116750 057100 477102 907192 097420 035041 777452 574825 976931 386263 125957 332363 663847 701625 > 5¹²⁸ [i]