MinT - Best Known (20, 20+17, s)-Nets in Base 64

Best Known (20, 20+17, s)-Nets in Base 64

(20, 20+17, 513)-Net over F₆₄ — Constructive and digital

Digital (20, 37, 513)-net over F₆₄, using

64¹ times duplication [i] based on digital (19, 36, 513)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64³⁶, 513, F₆₄, 17, 17) (dual of [(513, 17), 8685, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(64³⁶, 4105, F₆₄, 17) (dual of [4105, 4069, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(64³⁶, 4108, F₆₄, 17) (dual of [4108, 4072, 18]-code), using
      - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
        linear OA(64³³, 4097, F₆₄, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(64²⁵, 4097, F₆₄, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(64³, 11, F₆₄, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
        discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(20, 20+17, 516)-Net in Base 64 — Constructive

(20, 37, 516)-net in base 64, using

64¹ times duplication [i] based on (19, 36, 516)-net in base 64, using
- base change [i] based on digital (10, 27, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 9, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 18, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(20, 20+17, 2198)-Net over F₆₄ — Digital

Digital (20, 37, 2198)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁷, 2198, F₆₄, 17) (dual of [2198, 2161, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁷, 4110, F₆₄, 17) (dual of [4110, 4073, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(11) [i] based on
    1. linear OA(64³³, 4096, F₆₄, 17) (dual of [4096, 4063, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(64²³, 4096, F₆₄, 12) (dual of [4096, 4073, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    3. linear OA(64⁴, 14, F₆₄, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁴, 64, F₆₄, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
        Reedâ€“Solomon code RS(60,64) [i]

(20, 20+17, 8019719)-Net in Base 64 — Upper bound on s

There is no (20, 37, 8019720)-net in base 64, because

1 times m-reduction [i] would yield (20, 36, 8019720)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 105312 297737 015545 414403 703698 054029 689175 418789 417959 400234 979808 > 64³⁶ [i]