MinT - Best Known (63, 63+79, s)-Nets in Base 2

Best Known (63, 63+79, s)-Nets in Base 2

(63, 63+79, 43)-Net over F₂ — Constructive and digital

Digital (63, 142, 43)-net over F₂, using

t-expansion [i] based on digital (59, 142, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(63, 63+79, 44)-Net over F₂ — Digital

Digital (63, 142, 44)-net over F₂, using

t-expansion [i] based on digital (62, 142, 44)-net over F₂, using
- net from sequence [i] based on digital (62, 43)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 62 and N(F) ≥ 44, using
    - a function field from a table by Niederreiter and Xing [i]

(63, 63+79, 135)-Net over F₂ — Upper bound on s (digital)

There is no digital (63, 142, 136)-net over F₂, because

15 times m-reduction [i] would yield digital (63, 127, 136)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹²⁷, 136, F₂, 64) (dual of [136, 9, 65]-code), but
  - residual code [i] would yield linear OA(2⁶³, 71, F₂, 32) (dual of [71, 8, 33]-code), but
    - residual code [i] would yield linear OA(2³¹, 38, F₂, 16) (dual of [38, 7, 17]-code), but
      - residual code [i] would yield linear OA(2¹⁵, 21, F₂, 8) (dual of [21, 6, 9]-code), but
        residual code [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
        bound by Fontaine and Peterson [i]

(63, 63+79, 137)-Net in Base 2 — Upper bound on s

There is no (63, 142, 138)-net in base 2, because

1 times m-reduction [i] would yield (63, 141, 138)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 972226 293779 671711 309456 232088 676206 656624 > 2¹⁴¹ [i]