MinT - Best Known (111−84, 111, s)-Nets in Base 5

Best Known (111−84, 111, s)-Nets in Base 5

(111−84, 111, 51)-Net over F₅ — Constructive and digital

Digital (27, 111, 51)-net over F₅, using

t-expansion [i] based on digital (22, 111, 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]

(111−84, 111, 55)-Net over F₅ — Digital

Digital (27, 111, 55)-net over F₅, using

t-expansion [i] based on digital (23, 111, 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]

(111−84, 111, 258)-Net over F₅ — Upper bound on s (digital)

There is no digital (27, 111, 259)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹¹¹, 259, F₅, 84) (dual of [259, 148, 85]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹¹⁰, 144, S₅, 84), but
    - the linear programming bound shows that MÂ â‰¥ 91 409242 726417 624095 344105 505405 630376 260462 681874 772811 345983 107578 216930 733105 982653 796672 821044 921875 / 1070 506247 537363 410554 373824 > 5¹¹⁰ [i]
  2. linear OA(5¹⁴⁸, 259, F₅, 115) (dual of [259, 111, 116]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5¹⁴⁸, 250, F₅, 115) (dual of [250, 102, 116]-code), but
      - residual code [i] would yield OA(5³³, 134, S₅, 23), but
        the linear programming bound shows that MÂ â‰¥ 7 905984 850652 471068 879589 438438 415527 343750 / 66 073848 265952 909413 > 5³³ [i]

(111−84, 111, 260)-Net in Base 5 — Upper bound on s

There is no (27, 111, 261)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 411954 426073 740232 964072 329639 302240 836194 144317 280694 447655 717872 606672 021625 > 5¹¹¹ [i]