MinT - Best Known (253−91, 253, s)-Nets in Base 4

Best Known (253−91, 253, s)-Nets in Base 4

(253−91, 253, 200)-Net over F₄ — Constructive and digital

Digital (162, 253, 200)-net over F₄, using

t-expansion [i] based on digital (161, 253, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(253−91, 253, 208)-Net in Base 4 — Constructive

(162, 253, 208)-net in base 4, using

1 times m-reduction [i] based on (162, 254, 208)-net in base 4, using
- trace code for nets [i] based on (35, 127, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (35, 130, 104)-net in base 16, using
    - base change [i] based on digital (9, 104, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(253−91, 253, 517)-Net over F₄ — Digital

Digital (162, 253, 517)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−91, 253, 13785)-Net in Base 4 — Upper bound on s

There is no (162, 253, 13786)-net in base 4, because

1 times m-reduction [i] would yield (162, 252, 13786)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 443891 539557 702869 261209 928457 259560 310940 282176 898607 943951 759876 686456 184519 956199 795850 021784 228002 523128 009585 482111 437374 456266 323374 061737 890152 > 4²⁵² [i]