MinT - Best Known (77−39, 77, s)-Nets in Base 64

Best Known (77−39, 77, s)-Nets in Base 64

(77−39, 77, 513)-Net over F₆₄ — Constructive and digital

Digital (38, 77, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 77, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(77−39, 77, 1366)-Net over F₆₄ — Digital

Digital (38, 77, 1366)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁷, 1366, F₆₄, 3, 39) (dual of [(1366, 3), 4021, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(64⁷⁷, 4098, F₆₄, 39) (dual of [4098, 4021, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(37) [i] based on
    1. linear OA(64⁷⁷, 4096, F₆₄, 39) (dual of [4096, 4019, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    2. linear OA(64⁷⁵, 4096, F₆₄, 38) (dual of [4096, 4021, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(77−39, 77, 2111509)-Net in Base 64 — Upper bound on s

There is no (38, 77, 2111510)-net in base 64, because

1 times m-reduction [i] would yield (38, 76, 2111510)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 186072 364635 863226 195629 232565 949258 621618 557312 574222 486072 371579 875111 839352 946915 462486 121538 869732 817971 306610 412195 249408 056915 839278 > 64⁷⁶ [i]