MinT - Best Known (258−133, 258, s)-Nets in Base 2

Best Known (258−133, 258, s)-Nets in Base 2

(258−133, 258, 57)-Net over F₂ — Constructive and digital

Digital (125, 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]

(258−133, 258, 80)-Net over F₂ — Digital

Digital (125, 258, 80)-net over F₂, using

t-expansion [i] based on digital (121, 258, 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]

(258−133, 258, 259)-Net over F₂ — Upper bound on s (digital)

There is no digital (125, 258, 260)-net over F₂, because

5 times m-reduction [i] would yield digital (125, 253, 260)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵³, 260, F₂, 128) (dual of [260, 7, 129]-code), but
  - Griesmer bound [i]

(258−133, 258, 263)-Net in Base 2 — Upper bound on s

There is no (125, 258, 264)-net in base 2, because

1 times m-reduction [i] would yield (125, 257, 264)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁵⁷, 264, S₂, 132), but
  - adding a parity check bit [i] would yield OA(2²⁵⁸, 265, S₂, 133), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 44 464162 267129 419042 651258 243336 156615 655674 111605 976591 151712 259038 641781 735424 / 67 > 2²⁵⁸ [i]