MinT - Best Known (130, 260, s)-Nets in Base 2

Best Known (130, 260, s)-Nets in Base 2

(130, 260, 57)-Net over F₂ — Constructive and digital

Digital (130, 260, 57)-net over F₂, using

t-expansion [i] based on digital (110, 260, 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]

(130, 260, 81)-Net over F₂ — Digital

Digital (130, 260, 81)-net over F₂, using

t-expansion [i] based on digital (126, 260, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(130, 260, 272)-Net over F₂ — Upper bound on s (digital)

There is no digital (130, 260, 273)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁶⁰, 273, F₂, 130) (dual of [273, 13, 131]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(2²⁵⁹, 269, F₂, 130) (dual of [269, 10, 131]-code), but
    - residual code [i] would yield linear OA(2¹²⁹, 138, F₂, 65) (dual of [138, 9, 66]-code), but
      - 1 times truncation [i] would yield linear OA(2¹²⁸, 137, F₂, 64) (dual of [137, 9, 65]-code), but
        residual code [i] would yield linear OA(2⁶⁴, 72, F₂, 32) (dual of [72, 8, 33]-code), but
        adding a parity check bit [i] would yield linear OA(2⁶⁵, 73, F₂, 33) (dual of [73, 8, 34]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(2¹³, 273, S₂, 4), but
    - discarding factors would yield OA(2¹³, 128, S₂, 4), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 8257 > 2¹³ [i]

(130, 260, 286)-Net in Base 2 — Upper bound on s

There is no (130, 260, 287)-net in base 2, because

22 times m-reduction [i] would yield (130, 238, 287)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁸, 287, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 2 133520 632200 595168 112459 587583 515002 010526 895398 363193 388655 973626 286863 730797 742512 930816 / 4 628313 007677 734375 > 2²³⁸ [i]