MinT - Best Known (138, 238, s)-Nets in Base 4

Best Known (138, 238, s)-Nets in Base 4

(138, 238, 137)-Net over F₄ — Constructive and digital

Digital (138, 238, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 65, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 173, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(138, 238, 291)-Net over F₄ — Digital

Digital (138, 238, 291)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 238, 4727)-Net in Base 4 — Upper bound on s

There is no (138, 238, 4728)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 197013 742130 837589 616986 614075 589243 751816 429928 995152 880056 640828 055014 285899 415461 998451 130136 496430 215773 398186 773568 871119 992381 433900 481731 > 4²³⁸ [i]