MinT - Best Known (27, 27+86, s)-Nets in Base 5

Best Known (27, 27+86, s)-Nets in Base 5

(27, 27+86, 51)-Net over F₅ — Constructive and digital

Digital (27, 113, 51)-net over F₅, using

t-expansion [i] based on digital (22, 113, 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]

(27, 27+86, 55)-Net over F₅ — Digital

Digital (27, 113, 55)-net over F₅, using

t-expansion [i] based on digital (23, 113, 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]

(27, 27+86, 221)-Net over F₅ — Upper bound on s (digital)

There is no digital (27, 113, 222)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹¹³, 222, F₅, 86) (dual of [222, 109, 87]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹¹², 139, S₅, 86), but
    - the linear programming bound shows that MÂ â‰¥ 2 082589 264522 932824 279618 514424 429978 311367 037622 562595 373942 873067 107939 277775 585651 397705 078125 / 1 034045 105594 725224 > 5¹¹² [i]
  2. OA(5¹⁰⁹, 222, 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]

(27, 27+86, 259)-Net in Base 5 — Upper bound on s

There is no (27, 113, 260)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 10 656424 125764 291944 550724 538475 539309 877144 500147 737901 119508 871537 432603 089425 > 5¹¹³ [i]