MinT - Best Known (42−23, 42, s)-Nets in Base 64

Best Known (42−23, 42, s)-Nets in Base 64

(42−23, 42, 257)-Net over F₆₄ — Constructive and digital

Digital (19, 42, 257)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (7, 30, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(42−23, 42, 322)-Net in Base 64 — Constructive

(19, 42, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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]
2. (8, 31, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
    - base change [i] based on digital (0, 24, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(42−23, 42, 454)-Net over F₆₄ — Digital

Digital (19, 42, 454)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴², 454, F₆₄, 23) (dual of [454, 412, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁴², 455, F₆₄, 23) (dual of [455, 413, 24]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,4,7,…,67}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(42−23, 42, 513)-Net in Base 64

(19, 42, 513)-net in base 64, using

2 times m-reduction [i] based on (19, 44, 513)-net in base 64, using
- base change [i] based on digital (8, 33, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(42−23, 42, 420526)-Net in Base 64 — Upper bound on s

There is no (19, 42, 420527)-net in base 64, because

1 times m-reduction [i] would yield (19, 41, 420527)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 113 078970 073524 371835 088356 951539 741979 695288 981557 944350 916301 246230 802852 > 64⁴¹ [i]