MinT - Best Known (79−40, 79, s)-Nets in Base 64

Best Known (79−40, 79, s)-Nets in Base 64

(79−40, 79, 513)-Net over F₆₄ — Constructive and digital

Digital (39, 79, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 79, 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]

(79−40, 79, 1366)-Net over F₆₄ — Digital

Digital (39, 79, 1366)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁹, 1366, F₆₄, 3, 40) (dual of [(1366, 3), 4019, 41]-NRT-code), using
- OOA 3-folding [i] based on linear OA(64⁷⁹, 4098, F₆₄, 40) (dual of [4098, 4019, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(38) [i] based on
    1. linear OA(64⁷⁹, 4096, F₆₄, 40) (dual of [4096, 4017, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. 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]
    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]

(79−40, 79, 1796280)-Net in Base 64 — Upper bound on s

There is no (39, 79, 1796281)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 48777 480993 101599 110161 090483 703551 907358 042210 313289 608002 603238 793380 180904 152486 353731 923892 281677 423759 689781 747835 167088 745042 756256 281334 > 64⁷⁹ [i]