MinT - Best Known (82−23, 82, s)-Nets in Base 8

Best Known (82−23, 82, s)-Nets in Base 8

(82−23, 82, 484)-Net over F₈ — Constructive and digital

Digital (59, 82, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 22, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 11, 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. digital (37, 60, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 30, 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]

(82−23, 82, 576)-Net in Base 8 — Constructive

(59, 82, 576)-net in base 8, using

t-expansion [i] based on (57, 82, 576)-net in base 8, using
- 2 times m-reduction [i] based on (57, 84, 576)-net in base 8, using
  - trace code for nets [i] based on (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(82−23, 82, 3761)-Net over F₈ — Digital

Digital (59, 82, 3761)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸², 3761, F₈, 23) (dual of [3761, 3679, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸², 4106, F₈, 23) (dual of [4106, 4024, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(8⁸¹, 4097, F₈, 23) (dual of [4097, 4016, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(8⁷³, 4097, F₈, 21) (dual of [4097, 4024, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(82−23, 82, 3132863)-Net in Base 8 — Upper bound on s

There is no (59, 82, 3132864)-net in base 8, because

1 times m-reduction [i] would yield (59, 81, 3132864)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 134801 286621 422846 602548 090603 969913 655917 385280 966415 567151 839912 782713 > 8⁸¹ [i]