MinT - Best Known (30, 30+31, s)-Nets in Base 64

Best Known (30, 30+31, s)-Nets in Base 64

(30, 30+31, 513)-Net over F₆₄ — Constructive and digital

Digital (30, 61, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 61, 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]

(30, 30+31, 1302)-Net over F₆₄ — Digital

Digital (30, 61, 1302)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁶¹, 1302, F₆₄, 3, 31) (dual of [(1302, 3), 3845, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁶¹, 1366, F₆₄, 3, 31) (dual of [(1366, 3), 4037, 32]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(64⁶¹, 4098, F₆₄, 31) (dual of [4098, 4037, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(29) [i] based on
      1. linear OA(64⁶¹, 4096, F₆₄, 31) (dual of [4096, 4035, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(64⁵⁹, 4096, F₆₄, 30) (dual of [4096, 4037, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [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]

(30, 30+31, 1710582)-Net in Base 64 — Upper bound on s

There is no (30, 61, 1710583)-net in base 64, because

1 times m-reduction [i] would yield (30, 60, 1710583)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 348542 750401 298396 223596 140447 786238 091741 568510 756136 130761 025818 080529 629722 548330 071220 822160 367727 883248 > 64⁶⁰ [i]