MinT - Best Known (10, 10+10, s)-Nets in Base 2

Best Known (10, 10+10, s)-Nets in Base 2

(10, 10+10, 21)-Net over F₂ — Constructive and digital

Digital (10, 20, 21)-net over F₂, using

linear code embedding found in a computer search [i]

(10, 10+10, 31)-Net over F₂ — Upper bound on s (digital)

There is no digital (10, 20, 32)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁰, 32, F₂, 10) (dual of [32, 12, 11]-code), but
- adding a parity check bit [i] would yield linear OA(2²¹, 33, F₂, 11) (dual of [33, 12, 12]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(10, 10+10, 34)-Net in Base 2 — Upper bound on s

There is no (10, 20, 35)-net in base 2, because

extracting embedded OOA [i] would yield OOA(2²⁰, 35, S₂, 2, 10), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 1 375724 559937 241871 385458 860947 808750 108672 / 1 279869 485444 423328 561553 632013 861239 > 2²⁰ [i]