MinT - Best Known (248−87, 248, s)-Nets in Base 4

Best Known (248−87, 248, s)-Nets in Base 4

(248−87, 248, 200)-Net over F₄ — Constructive and digital

Digital (161, 248, 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]

(248−87, 248, 240)-Net in Base 4 — Constructive

(161, 248, 240)-net in base 4, using

2 times m-reduction [i] based on (161, 250, 240)-net in base 4, using
- trace code for nets [i] based on (36, 125, 120)-net in base 16, using
  - base change [i] based on digital (11, 100, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(248−87, 248, 555)-Net over F₄ — Digital

Digital (161, 248, 555)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−87, 248, 16134)-Net in Base 4 — Upper bound on s

There is no (161, 248, 16135)-net in base 4, because

1 times m-reduction [i] would yield (161, 247, 16135)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51280 113451 966730 054802 266769 732333 802962 847991 729407 703928 627186 769162 903705 716087 862439 658957 546725 282607 385703 851990 080292 452869 235077 335320 155392 > 4²⁴⁷ [i]