MinT - Best Known (102−65, 102, s)-Nets in Base 27

Best Known (102−65, 102, s)-Nets in Base 27

(102−65, 102, 114)-Net over F₂₇ — Constructive and digital

Digital (37, 102, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 102, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(102−65, 102, 172)-Net in Base 27 — Constructive

(37, 102, 172)-net in base 27, using

t-expansion [i] based on (34, 102, 172)-net in base 27, using
- 6 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
  - base change [i] based on digital (7, 81, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(102−65, 102, 244)-Net over F₂₇ — Digital

Digital (37, 102, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 102, 244)-net over F₂₇, using
- net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
    - Drinfeld modules of rank 1 [i]

(102−65, 102, 16188)-Net in Base 27 — Upper bound on s

There is no (37, 102, 16189)-net in base 27, because

1 times m-reduction [i] would yield (37, 101, 16189)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 701416 109578 162494 338008 592763 672242 381697 549484 786370 261381 299795 761640 325722 205329 861732 502113 122652 415121 666419 145898 352033 310674 295284 275329 > 27¹⁰¹ [i]