MinT - Best Known (93, 93+92, s)-Nets in Base 2

Best Known (93, 93+92, s)-Nets in Base 2

(93, 93+92, 53)-Net over F₂ — Constructive and digital

Digital (93, 185, 53)-net over F₂, using

t-expansion [i] based on digital (90, 185, 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]

(93, 93+92, 60)-Net over F₂ — Digital

Digital (93, 185, 60)-net over F₂, using

t-expansion [i] based on digital (92, 185, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(93, 93+92, 199)-Net over F₂ — Upper bound on s (digital)

There is no digital (93, 185, 200)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁵, 200, F₂, 92) (dual of [200, 15, 93]-code), but
- residual code [i] would yield linear OA(2⁹³, 107, F₂, 46) (dual of [107, 14, 47]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁹⁴, 108, F₂, 47) (dual of [108, 14, 48]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(93, 93+92, 230)-Net in Base 2 — Upper bound on s

There is no (93, 185, 231)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 54 336365 739379 104056 205188 581324 695237 078242 236625 463656 > 2¹⁸⁵ [i]