MinT - Best Known (73−37, 73, s)-Nets in Base 64

Best Known (73−37, 73, s)-Nets in Base 64

(73−37, 73, 513)-Net over F₆₄ — Constructive and digital

Digital (36, 73, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 73, 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]

(73−37, 73, 1366)-Net over F₆₄ — Digital

Digital (36, 73, 1366)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷³, 1366, F₆₄, 3, 37) (dual of [(1366, 3), 4025, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(64⁷³, 4098, F₆₄, 37) (dual of [4098, 4025, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(35) [i] based on
    1. linear OA(64⁷³, 4096, F₆₄, 37) (dual of [4096, 4023, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(64⁷¹, 4096, F₆₄, 36) (dual of [4096, 4025, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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]

(73−37, 73, 2011441)-Net in Base 64 — Upper bound on s

There is no (36, 73, 2011442)-net in base 64, because

1 times m-reduction [i] would yield (36, 72, 2011442)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11090 682437 102657 880761 392863 968841 065637 968111 338980 627691 336544 976051 191363 443416 543165 961359 594940 947688 854815 072021 872200 797296 > 64⁷² [i]