MinT - Best Known (62−46, 62, s)-Nets in Base 4

Best Known (62−46, 62, s)-Nets in Base 4

(62−46, 62, 33)-Net over F₄ — Constructive and digital

Digital (16, 62, 33)-net over F₄, using

t-expansion [i] based on digital (15, 62, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(62−46, 62, 36)-Net over F₄ — Digital

Digital (16, 62, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(62−46, 62, 74)-Net over F₄ — Upper bound on s (digital)

There is no digital (16, 62, 75)-net over F₄, because

extracting embedded orthogonal array [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]

(62−46, 62, 76)-Net in Base 4 — Upper bound on s

There is no (16, 62, 77)-net in base 4, because

1 times m-reduction [i] would yield (16, 61, 77)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶¹, 77, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 87771 116150 262149 392012 316054 470316 656445 882368 / 16226 709967 > 4⁶¹ [i]