MinT - Best Known (78, 78+21, s)-Nets in Base 27

Best Known (78, 78+21, s)-Nets in Base 27

(78, 78+21, 53230)-Net over F₂₇ — Constructive and digital

Digital (78, 99, 53230)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 86)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
    2. digital (2, 12, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (60, 81, 53144)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 53144, F₂₇, 21, 21) (dual of [(53144, 21), 1115943, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁸¹, 531441, F₂₇, 21) (dual of [531441, 531360, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(78, 78+21, 53261)-Net in Base 27 — Constructive

(78, 99, 53261)-net in base 27, using

27¹ times duplication [i] based on (77, 98, 53261)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (6, 16, 116)-net in base 27, using
    - base change [i] based on digital (2, 12, 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]
  2. digital (61, 82, 53145)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁸², 53145, F₂₇, 21, 21) (dual of [(53145, 21), 1115963, 22]-NRT-code), using
      - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁸², 531451, F₂₇, 21) (dual of [531451, 531369, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(27⁸¹, 531442, F₂₇, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(27⁷³, 531442, F₂₇, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 27⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(27¹, 9, F₂₇, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(78, 78+21, 3886732)-Net over F₂₇ — Digital

Digital (78, 99, 3886732)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+21, large)-Net in Base 27 — Upper bound on s

There is no (78, 99, large)-net in base 27, because

19 times m-reduction [i] would yield (78, 80, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (78, 78+21, s)-Nets in Base 27

(78, 78+21, 53230)-Net over F27 — Constructive and digital

(78, 78+21, 53261)-Net in Base 27 — Constructive

(78, 78+21, 3886732)-Net over F27 — Digital

(78, 78+21, large)-Net in Base 27 — Upper bound on s

(78, 78+21, 53230)-Net over F₂₇ — Constructive and digital

(78, 78+21, 3886732)-Net over F₂₇ — Digital