MinT - Best Known (260−138, 260, s)-Nets in Base 2

Best Known (260−138, 260, s)-Nets in Base 2

(260−138, 260, 57)-Net over F₂ — Constructive and digital

Digital (122, 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]

(260−138, 260, 80)-Net over F₂ — Digital

Digital (122, 260, 80)-net over F₂, using

t-expansion [i] based on digital (121, 260, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(260−138, 260, 255)-Net over F₂ — Upper bound on s (digital)

There is no digital (122, 260, 256)-net over F₂, because

10 times m-reduction [i] would yield digital (122, 250, 256)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁰, 256, F₂, 128) (dual of [256, 6, 129]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2²⁵¹, 257, F₂, 128) (dual of [257, 6, 129]-code), but
    - Griesmer bound [i]

(260−138, 260, 257)-Net in Base 2 — Upper bound on s

There is no (122, 260, 258)-net in base 2, because

8 times m-reduction [i] would yield (122, 252, 258)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁵², 258, S₂, 130), but
  - adding a parity check bit [i] would yield OA(2²⁵³, 259, S₂, 131), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 578960 446186 580977 117854 925043 439539 266349 923328 202820 197287 920039 565648 199680 / 33 > 2²⁵³ [i]