MinT - Best Known (253−128, 253, s)-Nets in Base 2

Best Known (253−128, 253, s)-Nets in Base 2

(253−128, 253, 57)-Net over F₂ — Constructive and digital

Digital (125, 253, 57)-net over F₂, using

t-expansion [i] based on digital (110, 253, 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]

(253−128, 253, 80)-Net over F₂ — Digital

Digital (125, 253, 80)-net over F₂, using

t-expansion [i] based on digital (121, 253, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(253−128, 253, 259)-Net over F₂ — Upper bound on s (digital)

There is no digital (125, 253, 260)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵³, 260, F₂, 128) (dual of [260, 7, 129]-code), but
- Griesmer bound [i]

(253−128, 253, 269)-Net in Base 2 — Upper bound on s

There is no (125, 253, 270)-net in base 2, because

20 times m-reduction [i] would yield (125, 233, 270)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³³, 270, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 1 805716 963838 191291 276449 773987 529933 551024 538218 073842 142319 599295 484110 749441 196032 / 124 860938 893295 > 2²³³ [i]