MinT - Best Known (207−147, 207, s)-Nets in Base 4

Best Known (207−147, 207, s)-Nets in Base 4

(207−147, 207, 66)-Net over F₄ — Constructive and digital

Digital (60, 207, 66)-net over F₄, using

t-expansion [i] based on digital (49, 207, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(207−147, 207, 91)-Net over F₄ — Digital

Digital (60, 207, 91)-net over F₄, using

t-expansion [i] based on digital (50, 207, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(207−147, 207, 408)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 207, 409)-net over F₄, because

3 times m-reduction [i] would yield digital (60, 204, 409)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁴, 409, F₄, 144) (dual of [409, 205, 145]-code), but
  - residual code [i] would yield OA(4⁶⁰, 264, S₄, 36), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1 330238 759813 440605 073984 426950 188656 > 4⁶⁰ [i]

(207−147, 207, 409)-Net in Base 4 — Upper bound on s

There is no (60, 207, 410)-net in base 4, because

1 times m-reduction [i] would yield (60, 206, 410)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11935 868273 383426 336644 432314 366002 490404 458533 034451 839115 882772 075293 948754 349636 648601 960310 139279 900577 247851 115629 790170 > 4²⁰⁶ [i]