MinT - Best Known (91, 91+103, s)-Nets in Base 2

Best Known (91, 91+103, s)-Nets in Base 2

(91, 91+103, 53)-Net over F₂ — Constructive and digital

Digital (91, 194, 53)-net over F₂, using

t-expansion [i] based on digital (90, 194, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(91, 91+103, 57)-Net over F₂ — Digital

Digital (91, 194, 57)-net over F₂, using

t-expansion [i] based on digital (83, 194, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(91, 91+103, 192)-Net over F₂ — Upper bound on s (digital)

There is no digital (91, 194, 193)-net over F₂, because

7 times m-reduction [i] would yield digital (91, 187, 193)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁷, 193, F₂, 96) (dual of [193, 6, 97]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹⁸⁸, 194, F₂, 96) (dual of [194, 6, 97]-code), but
    - Griesmer bound [i]

(91, 91+103, 194)-Net in Base 2 — Upper bound on s

There is no (91, 194, 195)-net in base 2, because

5 times m-reduction [i] would yield (91, 189, 195)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁸⁹, 195, S₂, 98), but
  - adding a parity check bit [i] would yield OA(2¹⁹⁰, 196, S₂, 99), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 50216 813883 093446 110686 315385 661331 328818 843555 712276 103168 / 25 > 2¹⁹⁰ [i]