MinT - Best Known (257−134, 257, s)-Nets in Base 2

Best Known (257−134, 257, s)-Nets in Base 2

(257−134, 257, 57)-Net over F₂ — Constructive and digital

Digital (123, 257, 57)-net over F₂, using

t-expansion [i] based on digital (110, 257, 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]

(257−134, 257, 80)-Net over F₂ — Digital

Digital (123, 257, 80)-net over F₂, using

t-expansion [i] based on digital (121, 257, 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]

(257−134, 257, 256)-Net over F₂ — Upper bound on s (digital)

There is no digital (123, 257, 257)-net over F₂, because

6 times m-reduction [i] would yield digital (123, 251, 257)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵¹, 257, F₂, 128) (dual of [257, 6, 129]-code), but
  - Griesmer bound [i]

(257−134, 257, 259)-Net in Base 2 — Upper bound on s

There is no (123, 257, 260)-net in base 2, because

4 times m-reduction [i] would yield (123, 253, 260)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁵³, 260, S₂, 130), but
  - adding a parity check bit [i] would yield OA(2²⁵⁴, 261, S₂, 131), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 463168 356949 264781 694283 940034 751631 413079 938662 562256 157830 336031 652518 559744 / 11 > 2²⁵⁴ [i]