MinT - Best Known (260−128, 260, s)-Nets in Base 2

Best Known (260−128, 260, s)-Nets in Base 2

(260−128, 260, 57)-Net over F₂ — Constructive and digital

Digital (132, 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]

(260−128, 260, 81)-Net over F₂ — Digital

Digital (132, 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]

(260−128, 260, 278)-Net over F₂ — Upper bound on s (digital)

There is no digital (132, 260, 279)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁶⁰, 279, F₂, 128) (dual of [279, 19, 129]-code), but
- residual code [i] would yield OA(2¹³², 150, S₂, 64), but
  - the linear programming bound shows that MÂ â‰¥ 7654 730789 395636 393332 135297 086550 086927 777792 / 1 285141 > 2¹³² [i]

(260−128, 260, 296)-Net in Base 2 — Upper bound on s

There is no (132, 260, 297)-net in base 2, because

20 times m-reduction [i] would yield (132, 240, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁰, 297, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 28020 398795 768772 015654 276848 365160 205592 563190 101250 876110 646415 347830 039205 258520 634937 311232 / 15527 302685 925212 739375 > 2²⁴⁰ [i]