MinT - Best Known (78−16, 78, s)-Nets in Base 3

Best Known (78−16, 78, s)-Nets in Base 3

(78−16, 78, 464)-Net over F₃ — Constructive and digital

Digital (62, 78, 464)-net over F₃, using

2 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]

(78−16, 78, 1260)-Net over F₃ — Digital

Digital (62, 78, 1260)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁸, 1260, F₃, 16) (dual of [1260, 1182, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁸, 2214, F₃, 16) (dual of [2214, 2136, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
    1. 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]
    2. linear OA(3⁵⁰, 2187, F₃, 11) (dual of [2187, 2137, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁷, 27, F₃, 4) (dual of [27, 20, 5]-code), using
      - an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(78−16, 78, 84441)-Net in Base 3 — Upper bound on s

There is no (62, 78, 84442)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 424648 763675 100761 900398 871566 683089 > 3⁷⁸ [i]