MinT - Best Known (43−21, 43, s)-Nets in Base 64

Best Known (43−21, 43, s)-Nets in Base 64

(43−21, 43, 410)-Net over F₆₄ — Constructive and digital

Digital (22, 43, 410)-net over F₆₄, using

64¹ times duplication [i] based on digital (21, 42, 410)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁴², 410, F₆₄, 21, 21) (dual of [(410, 21), 8568, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(64⁴², 4101, F₆₄, 21) (dual of [4101, 4059, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴², 4102, F₆₄, 21) (dual of [4102, 4060, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(64³⁷, 4097, F₆₄, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(43−21, 43, 514)-Net in Base 64 — Constructive

(22, 43, 514)-net in base 64, using

1 times m-reduction [i] based on (22, 44, 514)-net in base 64, using
- base change [i] based on digital (11, 33, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 11, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 22, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(43−21, 43, 1551)-Net over F₆₄ — Digital

Digital (22, 43, 1551)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴³, 1551, F₆₄, 2, 21) (dual of [(1551, 2), 3059, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁴³, 2052, F₆₄, 2, 21) (dual of [(2052, 2), 4061, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁴³, 4104, F₆₄, 21) (dual of [4104, 4061, 22]-code), using
    - constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
      1. linear OA(64⁴¹, 4096, F₆₄, 21) (dual of [4096, 4055, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(64³⁵, 4096, F₆₄, 18) (dual of [4096, 4061, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(43−21, 43, 2770708)-Net in Base 64 — Upper bound on s

There is no (22, 43, 2770709)-net in base 64, because

1 times m-reduction [i] would yield (22, 42, 2770709)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7237 005757 091166 537132 078512 920515 111794 714101 565355 866678 113579 551273 496180 > 64⁴² [i]