MinT - Best Known (103, 103+97, s)-Nets in Base 2

Best Known (103, 103+97, s)-Nets in Base 2

(103, 103+97, 55)-Net over F₂ — Constructive and digital

Digital (103, 200, 55)-net over F₂, using

t-expansion [i] based on digital (100, 200, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(103, 103+97, 65)-Net over F₂ — Digital

Digital (103, 200, 65)-net over F₂, using

t-expansion [i] based on digital (95, 200, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(103, 103+97, 230)-Net over F₂ — Upper bound on s (digital)

There is no digital (103, 200, 231)-net over F₂, because

1 times m-reduction [i] would yield digital (103, 199, 231)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁹, 231, F₂, 96) (dual of [231, 32, 97]-code), but
  - residual code [i] would yield OA(2¹⁰³, 134, S₂, 48), but
    - the linear programming bound shows that MÂ â‰¥ 14817 140394 002966 911489 108655 371959 926784 / 1391 278625 > 2¹⁰³ [i]

(103, 103+97, 266)-Net in Base 2 — Upper bound on s

There is no (103, 200, 267)-net in base 2, because

1 times m-reduction [i] would yield (103, 199, 267)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 856944 045510 773520 790281 237872 466464 530047 378199 716627 366960 > 2¹⁹⁹ [i]