MinT - Best Known (28, 28+89, s)-Nets in Base 5

Best Known (28, 28+89, s)-Nets in Base 5

(28, 28+89, 51)-Net over F₅ — Constructive and digital

Digital (28, 117, 51)-net over F₅, using

t-expansion [i] based on digital (22, 117, 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]

(28, 28+89, 55)-Net over F₅ — Digital

Digital (28, 117, 55)-net over F₅, using

t-expansion [i] based on digital (23, 117, 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]

(28, 28+89, 225)-Net over F₅ — Upper bound on s (digital)

There is no digital (28, 117, 226)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹¹⁷, 226, F₅, 89) (dual of [226, 109, 90]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹¹⁶, 143, S₅, 89), but
    - the linear programming bound shows that MÂ â‰¥ 2738 237588 593697 454668 762256 522048 887080 570758 264064 445890 203342 535984 543104 632393 806241 452693 939208 984375 / 2 228541 507840 976583 547528 > 5¹¹⁶ [i]
  2. OA(5¹⁰⁹, 226, S₅, 83), but
    - discarding factors would yield OA(5¹⁰⁹, 147, S₅, 83), but
      - the linear programming bound shows that MÂ â‰¥ 97832 005097 623965 961965 091455 398251 491670 818930 622769 438002 739592 061328 226246 796901 932611 945085 227489 471435 546875 / 5 803319 642355 292721 815985 390811 721467 > 5¹⁰⁹ [i]

(28, 28+89, 269)-Net in Base 5 — Upper bound on s

There is no (28, 117, 270)-net in base 5, because

1 times m-reduction [i] would yield (28, 116, 270)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1389 502776 644221 470896 356980 142959 445471 159743 772210 999294 708991 842325 507724 635585 > 5¹¹⁶ [i]