MinT - Best Known (122−35, 122, s)-Nets in Base 8

Best Known (122−35, 122, s)-Nets in Base 8

(122−35, 122, 514)-Net over F₈ — Constructive and digital

Digital (87, 122, 514)-net over F₈, using

8² times duplication [i] based on digital (85, 120, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 36, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 18, 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 (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(122−35, 122, 590)-Net in Base 8 — Constructive

(87, 122, 590)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [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]

(122−35, 122, 3831)-Net over F₈ — Digital

Digital (87, 122, 3831)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²², 3831, F₈, 35) (dual of [3831, 3709, 36]-code), using
- discarding factors / shortening the dual code 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]

(122−35, 122, 2746131)-Net in Base 8 — Upper bound on s

There is no (87, 122, 2746132)-net in base 8, because

1 times m-reduction [i] would yield (87, 121, 2746132)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 788415 401395 886071 589144 947417 091922 115226 336057 970776 697365 836657 274954 986138 297635 677978 251981 102020 996741 > 8¹²¹ [i]