MinT - Best Known (20, 38, s)-Nets in Base 64

Best Known (20, 38, s)-Nets in Base 64

(20, 38, 456)-Net over F₆₄ — Constructive and digital

Digital (20, 38, 456)-net over F₆₄, using

64¹ times duplication [i] based on digital (19, 37, 456)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64³⁷, 456, F₆₄, 18, 18) (dual of [(456, 18), 8171, 19]-NRT-code), using
  - OA 9-folding and stacking [i] based on linear OA(64³⁷, 4104, F₆₄, 18) (dual of [4104, 4067, 19]-code), using
    - constructionÂ X applied to C_e(17) âŠ‚ C_e(14) [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(64²⁹, 4096, F₆₄, 15) (dual of [4096, 4067, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [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]

(20, 38, 515)-Net in Base 64 — Constructive

(20, 38, 515)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (3, 12, 257)-net in base 64, using
  - base change [i] based on digital (0, 9, 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. (8, 26, 258)-net in base 64, using
  - 2 times m-reduction [i] based on (8, 28, 258)-net in base 64, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(20, 38, 2053)-Net over F₆₄ — Digital

Digital (20, 38, 2053)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64³⁸, 2053, F₆₄, 2, 18) (dual of [(2053, 2), 4068, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64³⁸, 4106, F₆₄, 18) (dual of [4106, 4068, 19]-code), using
  - discarding factors / shortening the dual code based on linear OA(64³⁸, 4107, F₆₄, 18) (dual of [4107, 4069, 19]-code), using
    - constructionÂ X applied to C_e(17) âŠ‚ C_e(13) [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(64²⁷, 4096, F₆₄, 14) (dual of [4096, 4069, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      3. 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, 38, 2782937)-Net in Base 64 — Upper bound on s

There is no (20, 38, 2782938)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 431 359641 449166 567049 213075 571857 840451 329637 561039 732022 299235 206425 > 64³⁸ [i]