MinT - Best Known (103, 202, s)-Nets in Base 2

Best Known (103, 202, s)-Nets in Base 2

(103, 202, 55)-Net over F₂ — Constructive and digital

Digital (103, 202, 55)-net over F₂, using

t-expansion [i] based on digital (100, 202, 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, 202, 65)-Net over F₂ — Digital

Digital (103, 202, 65)-net over F₂, using

t-expansion [i] based on digital (95, 202, 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, 202, 228)-Net over F₂ — Upper bound on s (digital)

There is no digital (103, 202, 229)-net over F₂, because

1 times m-reduction [i] would yield digital (103, 201, 229)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁰¹, 229, F₂, 98) (dual of [229, 28, 99]-code), but
  - residual code [i] would yield OA(2¹⁰³, 130, S₂, 49), but
    - 1 times truncation [i] would yield OA(2¹⁰², 129, S₂, 48), but
      - the linear programming bound shows that MÂ â‰¥ 4505 293676 588367 515344 369298 908326 658048 / 781 535645 > 2¹⁰² [i]

(103, 202, 261)-Net in Base 2 — Upper bound on s

There is no (103, 202, 262)-net in base 2, because

1 times m-reduction [i] would yield (103, 201, 262)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 295286 453629 242357 756343 045559 228046 328668 373199 526249 194864 > 2²⁰¹ [i]