MinT - Best Known (67−34, 67, s)-Nets in Base 64

Best Known (67−34, 67, s)-Nets in Base 64

(67−34, 67, 513)-Net over F₆₄ — Constructive and digital

Digital (33, 67, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 67, 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]

(67−34, 67, 1349)-Net over F₆₄ — Digital

Digital (33, 67, 1349)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁶⁷, 1349, F₆₄, 3, 34) (dual of [(1349, 3), 3980, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁶⁷, 1366, F₆₄, 3, 34) (dual of [(1366, 3), 4031, 35]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(64⁶⁷, 4098, F₆₄, 34) (dual of [4098, 4031, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
      1. linear OA(64⁶⁷, 4096, F₆₄, 34) (dual of [4096, 4029, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(64⁶⁵, 4096, F₆₄, 33) (dual of [4096, 4031, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [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]

(67−34, 67, 1496498)-Net in Base 64 — Upper bound on s

There is no (33, 67, 1496499)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 10 329063 593833 899303 650360 649283 287942 806630 708421 493967 708892 579079 709529 089249 455221 217982 964073 773059 281894 390567 708930 > 64⁶⁷ [i]