MinT - Best Known (257−107, 257, s)-Nets in Base 4

Best Known (257−107, 257, s)-Nets in Base 4

(257−107, 257, 138)-Net over F₄ — Constructive and digital

Digital (150, 257, 138)-net over F₄, using

t-expansion [i] based on digital (149, 257, 138)-net over F₄, using
- 2 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 76, 34)-net over F₄, using
      - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 183, 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]

(257−107, 257, 139)-Net in Base 4 — Constructive

(150, 257, 139)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (24, 77, 35)-net in base 4, using
  - net from sequence [i] based on (24, 34)-sequence in base 4, using
    - base expansion [i] based on digital (48, 34)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (73, 180, 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]

(257−107, 257, 321)-Net over F₄ — Digital

Digital (150, 257, 321)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−107, 257, 5512)-Net in Base 4 — Upper bound on s

There is no (150, 257, 5513)-net in base 4, because

1 times m-reduction [i] would yield (150, 256, 5513)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13524 494447 654522 028355 552662 169612 523993 069833 051451 188805 585028 697031 302929 775588 723219 227839 876576 901138 584742 303436 929254 164978 454357 311504 576783 797760 > 4²⁵⁶ [i]