MinT - Best Known (123, 123+124, s)-Nets in Base 2

Best Known (123, 123+124, s)-Nets in Base 2

(123, 123+124, 57)-Net over F₂ — Constructive and digital

Digital (123, 247, 57)-net over F₂, using

t-expansion [i] based on digital (110, 247, 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]

(123, 123+124, 80)-Net over F₂ — Digital

Digital (123, 247, 80)-net over F₂, using

t-expansion [i] based on digital (121, 247, 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]

(123, 123+124, 258)-Net over F₂ — Upper bound on s (digital)

There is no digital (123, 247, 259)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁷, 259, F₂, 124) (dual of [259, 12, 125]-code), but
- residual code [i] would yield OA(2¹²³, 134, S₂, 62), but
  - the linear programming bound shows that MÂ â‰¥ 2988 104534 524490 882287 758271 510214 606848 / 247 > 2¹²³ [i]

(123, 123+124, 262)-Net in Base 2 — Upper bound on s

There is no (123, 247, 263)-net in base 2, because

14 times m-reduction [i] would yield (123, 233, 263)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³³, 263, S₂, 110), but
  - the linear programming bound shows that MÂ â‰¥ 208 361079 889821 744668 890387 660731 037374 858559 867656 192612 204735 454426 713127 124992 / 13955 210325 > 2²³³ [i]