MinT - Best Known (60, 60+77, s)-Nets in Base 2

Best Known (60, 60+77, s)-Nets in Base 2

(60, 60+77, 43)-Net over F₂ — Constructive and digital

Digital (60, 137, 43)-net over F₂, using

t-expansion [i] based on digital (59, 137, 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]

(60, 60+77, 129)-Net over F₂ — Upper bound on s (digital)

There is no digital (60, 137, 130)-net over F₂, because

13 times m-reduction [i] would yield digital (60, 124, 130)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹²⁴, 130, F₂, 64) (dual of [130, 6, 65]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹²⁵, 131, F₂, 64) (dual of [131, 6, 65]-code), but
    - Griesmer bound [i]

(60, 60+77, 130)-Net in Base 2 — Upper bound on s

There is no (60, 137, 131)-net in base 2, because

1 times m-reduction [i] would yield (60, 136, 131)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 108208 456442 601657 925151 556934 079556 502368 > 2¹³⁶ [i]