MinT - Best Known (256−136, 256, s)-Nets in Base 2

Best Known (256−136, 256, s)-Nets in Base 2

(256−136, 256, 57)-Net over F₂ — Constructive and digital

Digital (120, 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]

(256−136, 256, 73)-Net over F₂ — Digital

Digital (120, 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]

(256−136, 256, 252)-Net over F₂ — Upper bound on s (digital)

There is no digital (120, 256, 253)-net over F₂, because

16 times m-reduction [i] would yield digital (120, 240, 253)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁰, 253, F₂, 120) (dual of [253, 13, 121]-code), but
  - residual code [i] would yield OA(2¹²⁰, 132, S₂, 60), but
    - the linear programming bound shows that MÂ â‰¥ 16503 694795 665515 477973 668460 440758 255616 / 10323 > 2¹²⁰ [i]

(256−136, 256, 253)-Net in Base 2 — Upper bound on s

There is no (120, 256, 254)-net in base 2, because

8 times m-reduction [i] would yield (120, 248, 254)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁸, 254, S₂, 128), but
  - adding a parity check bit [i] would yield OA(2²⁴⁹, 255, S₂, 129), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 14474 011154 664524 427946 373126 085988 481658 748083 205070 504932 198000 989141 204992 / 13 > 2²⁴⁹ [i]