MinT - Best Known (30−14, 30, s)-Nets in Base 2

Best Known (30−14, 30, s)-Nets in Base 2

(30−14, 30, 18)-Net over F₂ — Constructive and digital

Digital (16, 30, 18)-net over F₂, using

trace code for nets [i] based on digital (1, 15, 9)-net over F₄, using
- net from sequence [i] based on digital (1, 8)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
    - the Hermitian function field over F₄ [i]

(30−14, 30, 47)-Net over F₂ — Upper bound on s (digital)

There is no digital (16, 30, 48)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2³⁰, 48, F₂, 14) (dual of [48, 18, 15]-code), but
- adding a parity check bit [i] would yield linear OA(2³¹, 49, F₂, 15) (dual of [49, 18, 16]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(30−14, 30, 56)-Net in Base 2 — Upper bound on s

There is no (16, 30, 57)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1126 764544 > 2³⁰ [i]