MinT - Best Known (36, 36+18, s)-Nets in Base 4

Best Known (36, 36+18, s)-Nets in Base 4

(36, 36+18, 195)-Net over F₄ — Constructive and digital

Digital (36, 54, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 18, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(36, 36+18, 213)-Net over F₄ — Digital

Digital (36, 54, 213)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁴, 213, F₄, 18) (dual of [213, 159, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁴, 255, F₄, 18) (dual of [255, 201, 19]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(36, 36+18, 5655)-Net in Base 4 — Upper bound on s

There is no (36, 54, 5656)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 324 896444 908615 566273 442635 767962 > 4⁵⁴ [i]