MinT - Best Known (231−119, 231, s)-Nets in Base 2

Best Known (231−119, 231, s)-Nets in Base 2

(231−119, 231, 57)-Net over F₂ — Constructive and digital

Digital (112, 231, 57)-net over F₂, using

t-expansion [i] based on digital (110, 231, 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]

(231−119, 231, 72)-Net over F₂ — Digital

Digital (112, 231, 72)-net over F₂, using

t-expansion [i] based on digital (110, 231, 72)-net over F₂, using
- net from sequence [i] based on digital (110, 71)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 110 and N(F) ≥ 72, using
    - Drinfeld modules of rank 1 [i]

(231−119, 231, 235)-Net over F₂ — Upper bound on s (digital)

There is no digital (112, 231, 236)-net over F₂, because

7 times m-reduction [i] would yield digital (112, 224, 236)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²²⁴, 236, F₂, 112) (dual of [236, 12, 113]-code), but
  - residual code [i] would yield linear OA(2¹¹², 123, F₂, 56) (dual of [123, 11, 57]-code), but
    - residual code [i] would yield linear OA(2⁵⁶, 66, F₂, 28) (dual of [66, 10, 29]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁵⁷, 67, F₂, 29) (dual of [67, 10, 30]-code), but
        â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(231−119, 231, 236)-Net in Base 2 — Upper bound on s

There is no (112, 231, 237)-net in base 2, because

13 times m-reduction [i] would yield (112, 218, 237)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁸, 237, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 59311 882667 731407 548267 433191 443187 482001 717729 589259 458427 942548 275200 / 110349 > 2²¹⁸ [i]