MinT - Best Known (257−103, 257, s)-Nets in Base 4

Best Known (257−103, 257, s)-Nets in Base 4

(257−103, 257, 138)-Net over F₄ — Constructive and digital

Digital (154, 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]

(257−103, 257, 147)-Net in Base 4 — Constructive

(154, 257, 147)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (30, 81, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
2. digital (73, 176, 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]

(257−103, 257, 363)-Net over F₄ — Digital

Digital (154, 257, 363)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−103, 257, 6921)-Net in Base 4 — Upper bound on s

There is no (154, 257, 6922)-net in base 4, because

1 times m-reduction [i] would yield (154, 256, 6922)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13428 532344 194403 032621 967472 684265 213825 283359 326503 949843 768684 000973 274905 220376 806723 980230 989029 214763 072428 712228 649704 088298 701095 770046 158282 399040 > 4²⁵⁶ [i]