MinT - Best Known (123−35, 123, s)-Nets in Base 8

Best Known (123−35, 123, s)-Nets in Base 8

(123−35, 123, 514)-Net over F₈ — Constructive and digital

Digital (88, 123, 514)-net over F₈, using

1 times m-reduction [i] based on digital (88, 124, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 38, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 19, 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 (50, 86, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 43, 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]

(123−35, 123, 593)-Net in Base 8 — Constructive

(88, 123, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (69, 104, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 52, 288)-net in base 64, using
    - 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
      - base change [i] based on digital (9, 48, 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]

(123−35, 123, 4082)-Net over F₈ — Digital

Digital (88, 123, 4082)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²³, 4082, F₈, 35) (dual of [4082, 3959, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹²³, 4107, F₈, 35) (dual of [4107, 3984, 36]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹²², 4106, F₈, 35) (dual of [4106, 3984, 36]-code), using
    - constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(8¹²¹, 4097, F₈, 35) (dual of [4097, 3976, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
      2. linear OA(8¹¹³, 4097, F₈, 33) (dual of [4097, 3984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (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]

(123−35, 123, 3103447)-Net in Base 8 — Upper bound on s

There is no (88, 123, 3103448)-net in base 8, because

1 times m-reduction [i] would yield (88, 122, 3103448)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 306933 357489 460203 584530 700582 911644 691703 628959 088844 740362 210440 021362 434556 622686 406274 128863 401600 172417 > 8¹²² [i]