MinT - Best Known (60−16, 60, s)-Nets in Base 8

Best Known (60−16, 60, s)-Nets in Base 8

(60−16, 60, 513)-Net over F₈ — Constructive and digital

Digital (44, 60, 513)-net over F₈, using

8¹ times duplication [i] based on digital (43, 59, 513)-net over F₈, using
- t-expansion [i] based on digital (42, 59, 513)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁹, 513, F₈, 17, 17) (dual of [(513, 17), 8662, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(8⁵⁹, 4105, F₈, 17) (dual of [4105, 4046, 18]-code), using
      - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
        linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(8⁴⁹, 4096, F₈, 14) (dual of [4096, 4047, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(8², 9, F₈, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
        extended Reedâ€“Solomon code RS_e(7,8) [i]
        Hamming code H(2,8) [i]

(60−16, 60, 644)-Net in Base 8 — Constructive

(44, 60, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (8, 16, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. (28, 44, 514)-net in base 8, using
  - base change [i] based on digital (17, 33, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (17, 34, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 17, 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]

(60−16, 60, 4261)-Net over F₈ — Digital

Digital (44, 60, 4261)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁰, 4261, F₈, 16) (dual of [4261, 4201, 17]-code), using
- 161 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 35 times 0, 1, 117 times 0) [i] based on linear OA(8⁵⁶, 4096, F₈, 16) (dual of [4096, 4040, 17]-code), using
  - 1 times truncation [i] based on linear OA(8⁵⁷, 4097, F₈, 17) (dual of [4097, 4040, 18]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(60−16, 60, 3189820)-Net in Base 8 — Upper bound on s

There is no (44, 60, 3189821)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 532496 024746 931559 231462 625564 117792 774879 936231 730157 > 8⁶⁰ [i]