MinT - Best Known (30, 30+113, s)-Nets in Base 5

Best Known (30, 30+113, s)-Nets in Base 5

(30, 30+113, 51)-Net over F₅ — Constructive and digital

Digital (30, 143, 51)-net over F₅, using

t-expansion [i] based on digital (22, 143, 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, 30+113, 58)-Net over F₅ — Digital

Digital (30, 143, 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, 30+113, 163)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 143, 164)-net over F₅, because

4 times m-reduction [i] would yield digital (30, 139, 164)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹³⁹, 164, F₅, 109) (dual of [164, 25, 110]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(5¹³⁸, 148, S₅, 109), but
      - the linear programming bound shows that MÂ â‰¥ 86 158555 819318 512642 331328 682663 835348 540649 443889 196923 328992 021307 026760 723601 910285 651683 807373 046875 / 29 425011 > 5¹³⁸ [i]
    2. OA(5²⁵, 164, S₅, 16), but
      - discarding factors would yield OA(5²⁵, 148, S₅, 16), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 313081 833470 800945 > 5²⁵ [i]

(30, 30+113, 281)-Net in Base 5 — Upper bound on s

There is no (30, 143, 282)-net in base 5, because

1 times m-reduction [i] would yield (30, 142, 282)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1912 164913 274706 035417 052715 707460 137283 821739 068677 742260 002416 024280 805445 040204 196972 723367 935105 > 5¹⁴² [i]