MinT - Best Known (258−140, 258, s)-Nets in Base 2

Best Known (258−140, 258, s)-Nets in Base 2

(258−140, 258, 57)-Net over F₂ — Constructive and digital

Digital (118, 258, 57)-net over F₂, using

t-expansion [i] based on digital (110, 258, 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]

(258−140, 258, 73)-Net over F₂ — Digital

Digital (118, 258, 73)-net over F₂, using

t-expansion [i] based on digital (114, 258, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(258−140, 258, 247)-Net over F₂ — Upper bound on s (digital)

There is no digital (118, 258, 248)-net over F₂, because

20 times m-reduction [i] would yield digital (118, 238, 248)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁸, 248, F₂, 120) (dual of [248, 10, 121]-code), but
  - residual code [i] would yield linear OA(2¹¹⁸, 127, F₂, 60) (dual of [127, 9, 61]-code), but
    - residual code [i] would yield linear OA(2⁵⁸, 66, F₂, 30) (dual of [66, 8, 31]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁵⁹, 67, F₂, 31) (dual of [67, 8, 32]-code), but
        â€œDHMâ€ bound on codes from Brouwerâ€™s database [i]

(258−140, 258, 248)-Net in Base 2 — Upper bound on s

There is no (118, 258, 249)-net in base 2, because

16 times m-reduction [i] would yield (118, 242, 249)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴², 249, S₂, 124), but
  - adding a parity check bit [i] would yield OA(2²⁴³, 250, S₂, 125), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 904 625697 166532 776746 648320 380374 280103 671755 200316 906558 262375 061821 325312 / 63 > 2²⁴³ [i]