MinT - Best Known (258−143, 258, s)-Nets in Base 2

Best Known (258−143, 258, s)-Nets in Base 2

(258−143, 258, 57)-Net over F₂ — Constructive and digital

Digital (115, 258, 57)-net over F₂, using

t-expansion [i] based on digital (110, 258, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(258−143, 258, 73)-Net over F₂ — Digital

Digital (115, 258, 73)-net over F₂, using

t-expansion [i] based on digital (114, 258, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(258−143, 258, 241)-Net over F₂ — Upper bound on s (digital)

There is no digital (115, 258, 242)-net over F₂, because

23 times m-reduction [i] would yield digital (115, 235, 242)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁵, 242, F₂, 120) (dual of [242, 7, 121]-code), but
  - Griesmer bound [i]

(258−143, 258, 242)-Net in Base 2 — Upper bound on s

There is no (115, 258, 243)-net in base 2, because

1 times m-reduction [i] would yield (115, 257, 243)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 269682 517797 887243 468770 171118 277486 051128 950168 649335 116117 389232 038136 000088 > 2²⁵⁷ [i]