MinT - Best Known (248−133, 248, s)-Nets in Base 2

Best Known (248−133, 248, s)-Nets in Base 2

(248−133, 248, 57)-Net over F₂ — Constructive and digital

Digital (115, 248, 57)-net over F₂, using

t-expansion [i] based on digital (110, 248, 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]

(248−133, 248, 73)-Net over F₂ — Digital

Digital (115, 248, 73)-net over F₂, using

t-expansion [i] based on digital (114, 248, 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]

(248−133, 248, 241)-Net over F₂ — Upper bound on s (digital)

There is no digital (115, 248, 242)-net over F₂, because

13 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]

(248−133, 248, 242)-Net in Base 2 — Upper bound on s

There is no (115, 248, 243)-net in base 2, because

25 times m-reduction [i] would yield (115, 223, 243)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²³, 243, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 110427 941548 649020 598956 093796 432407 239217 743554 726184 882600 387580 788736 / 5797 > 2²²³ [i]