MinT - Best Known (116−88, 116, s)-Nets in Base 5

Best Known (116−88, 116, s)-Nets in Base 5

(116−88, 116, 51)-Net over F₅ — Constructive and digital

Digital (28, 116, 51)-net over F₅, using

t-expansion [i] based on digital (22, 116, 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]

(116−88, 116, 55)-Net over F₅ — Digital

Digital (28, 116, 55)-net over F₅, using

t-expansion [i] based on digital (23, 116, 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]

(116−88, 116, 251)-Net over F₅ — Upper bound on s (digital)

There is no digital (28, 116, 252)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹¹⁶, 252, F₅, 88) (dual of [252, 136, 89]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹¹⁵, 147, S₅, 88), but
    - the linear programming bound shows that MÂ â‰¥ 2106 427198 388426 824206 051416 107971 920370 889275 492662 277335 854828 334837 929304 512726 957909 762859 344482 421875 / 6 144065 730517 820384 976983 > 5¹¹⁵ [i]
  2. linear OA(5¹³⁶, 252, F₅, 105) (dual of [252, 116, 106]-code), but
    - residual code [i] would yield OA(5³¹, 146, S₅, 21), but
      - 1 times truncation [i] would yield OA(5³⁰, 145, S₅, 20), but
        the linear programming bound shows that MÂ â‰¥ 34078 027381 828650 342719 814777 374267 578125 / 36 570326 005604 057357 > 5³⁰ [i]

(116−88, 116, 269)-Net in Base 5 — Upper bound on s

There is no (28, 116, 270)-net in base 5, because

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]