MinT - Best Known (137, 137+79, s)-Nets in Base 4

Best Known (137, 137+79, s)-Nets in Base 4

(137, 137+79, 151)-Net over F₄ — Constructive and digital

Digital (137, 216, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 46, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (91, 170, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 85, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(137, 137+79, 196)-Net in Base 4 — Constructive

(137, 216, 196)-net in base 4, using

trace code for nets [i] based on (29, 108, 98)-net in base 16, using
- 2 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
  - base change [i] based on digital (7, 88, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(137, 137+79, 424)-Net over F₄ — Digital

Digital (137, 216, 424)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+79, 10667)-Net in Base 4 — Upper bound on s

There is no (137, 216, 10668)-net in base 4, because

1 times m-reduction [i] would yield (137, 215, 10668)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2780 053729 133496 101124 892239 054039 192554 216165 383471 656006 731039 196648 651461 931673 405774 807377 902589 470555 154806 125056 043882 587430 > 4²¹⁵ [i]