MinT - Best Known (256−139, 256, s)-Nets in Base 2

Best Known (256−139, 256, s)-Nets in Base 2

(256−139, 256, 57)-Net over F₂ — Constructive and digital

Digital (117, 256, 57)-net over F₂, using

t-expansion [i] based on digital (110, 256, 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]

(256−139, 256, 73)-Net over F₂ — Digital

Digital (117, 256, 73)-net over F₂, using

t-expansion [i] based on digital (114, 256, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(256−139, 256, 246)-Net in Base 2 — Upper bound on s

There is no (117, 256, 247)-net in base 2, because

15 times m-reduction [i] would yield (117, 241, 247)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴¹, 247, S₂, 124), but
  - adding a parity check bit [i] would yield OA(2²⁴², 248, S₂, 125), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 452 312848 583266 388373 324160 190187 140051 835877 600158 453279 131187 530910 662656 / 63 > 2²⁴² [i]