MinT - Best Known (63, 63+16, s)-Nets in Base 3

Best Known (63, 63+16, s)-Nets in Base 3

(63, 63+16, 464)-Net over F₃ — Constructive and digital

Digital (63, 79, 464)-net over F₃, using

t-expansion [i] based on digital (62, 79, 464)-net over F₃, using
- 1 times m-reduction [i] based on digital (62, 80, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 20, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(63, 63+16, 1363)-Net over F₃ — Digital

Digital (63, 79, 1363)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁹, 1363, F₃, 16) (dual of [1363, 1284, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁹, 2196, F₃, 16) (dual of [2196, 2117, 17]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3⁸, 9, F₃, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
      - dual of repetition code with length 9 [i]
    2. linear OA(3⁷¹, 2187, F₃, 16) (dual of [2187, 2116, 17]-code), using
      - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(63, 63+16, 96872)-Net in Base 3 — Upper bound on s

There is no (63, 79, 96873)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49 273158 857994 041206 326568 619866 770081 > 3⁷⁹ [i]