MinT - Best Known (21, 39, s)-Nets in Base 64

Best Known (21, 39, s)-Nets in Base 64

(21, 39, 456)-Net over F₆₄ — Constructive and digital

Digital (21, 39, 456)-net over F₆₄, using

1 times m-reduction [i] based on digital (21, 40, 456)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁴⁰, 456, F₆₄, 19, 19) (dual of [(456, 19), 8624, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(64⁴⁰, 4105, F₆₄, 19) (dual of [4105, 4065, 20]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴⁰, 4108, F₆₄, 19) (dual of [4108, 4068, 20]-code), using
      - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
        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(64²⁹, 4097, F₆₄, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (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]

(21, 39, 516)-Net in Base 64 — Constructive

(21, 39, 516)-net in base 64, using

1 times m-reduction [i] based on (21, 40, 516)-net in base 64, using
- base change [i] based on digital (11, 30, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 10, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 20, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(21, 39, 2096)-Net over F₆₄ — Digital

Digital (21, 39, 2096)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁹, 2096, F₆₄, 18) (dual of [2096, 2057, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁹, 4110, F₆₄, 18) (dual of [4110, 4071, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(12) [i] based on
    1. 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]
    2. linear OA(64²⁵, 4096, F₆₄, 13) (dual of [4096, 4071, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(21, 39, 4417640)-Net in Base 64 — Upper bound on s

There is no (21, 39, 4417641)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 27607 009050 468317 128524 662534 567619 189915 270723 810831 221917 334668 951648 > 64³⁹ [i]