MinT - Best Known (72−19, 72, s)-Nets in Base 8

Best Known (72−19, 72, s)-Nets in Base 8

(72−19, 72, 514)-Net over F₈ — Constructive and digital

Digital (53, 72, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 10, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (33, 52, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 26, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(72−19, 72, 674)-Net in Base 8 — Constructive

(53, 72, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 10, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. (33, 52, 514)-net in base 8, using
  - base change [i] based on digital (20, 39, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (20, 40, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 20, 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]

(72−19, 72, 4571)-Net over F₈ — Digital

Digital (53, 72, 4571)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷², 4571, F₈, 19) (dual of [4571, 4499, 20]-code), using
- 459 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 16 times 0, 1, 47 times 0, 1, 123 times 0, 1, 264 times 0) [i] based on linear OA(8⁶⁶, 4106, F₈, 19) (dual of [4106, 4040, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(8⁶⁵, 4097, F₈, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. 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]
    3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(72−19, 72, 7889140)-Net in Base 8 — Upper bound on s

There is no (53, 72, 7889141)-net in base 8, because

1 times m-reduction [i] would yield (53, 71, 7889141)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 046545 910435 569517 854590 425182 276445 552686 790684 879917 092424 > 8⁷¹ [i]