MinT - Best Known (253−191, 253, s)-Nets in Base 4

Best Known (253−191, 253, s)-Nets in Base 4

(253−191, 253, 66)-Net over F₄ — Constructive and digital

Digital (62, 253, 66)-net over F₄, using

t-expansion [i] based on digital (49, 253, 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]

(253−191, 253, 99)-Net over F₄ — Digital

Digital (62, 253, 99)-net over F₄, using

t-expansion [i] based on digital (61, 253, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(253−191, 253, 259)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 253, 260)-net over F₄, because

7 times m-reduction [i] would yield digital (62, 246, 260)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁶, 260, F₄, 184) (dual of [260, 14, 185]-code), but
  - residual code [i] would yield linear OA(4⁶², 75, F₄, 46) (dual of [75, 13, 47]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(253−191, 253, 402)-Net in Base 4 — Upper bound on s

There is no (62, 253, 403)-net in base 4, because

1 times m-reduction [i] would yield (62, 252, 403)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 60 404686 804467 248482 146051 620477 777431 976079 989125 508189 874913 166435 187265 961088 924065 456324 175551 452034 365714 212935 969980 380831 173812 671105 884167 825408 > 4²⁵² [i]