MinT - Best Known (109−30, 109, s)-Nets in Base 8

Best Known (109−30, 109, s)-Nets in Base 8

(109−30, 109, 514)-Net over F₈ — Constructive and digital

Digital (79, 109, 514)-net over F₈, using

8¹ times duplication [i] based on digital (78, 108, 514)-net over F₈, using
- t-expansion [i] based on digital (77, 108, 514)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (17, 32, 160)-net over F₈, using
      - trace code for nets [i] based on digital (1, 16, 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 (45, 76, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 38, 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]

(109−30, 109, 593)-Net in Base 8 — Constructive

(79, 109, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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. (62, 92, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 46, 288)-net in base 64, using
    - 3 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
      - base change [i] based on digital (9, 42, 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]

(109−30, 109, 4268)-Net over F₈ — Digital

Digital (79, 109, 4268)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁹, 4268, F₈, 30) (dual of [4268, 4159, 31]-code), using
- 164 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 38 times 0, 1, 117 times 0) [i] based on linear OA(8¹⁰⁵, 4100, F₈, 30) (dual of [4100, 3995, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(8¹⁰⁵, 4096, F₈, 30) (dual of [4096, 3991, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(109−30, 109, 3350590)-Net in Base 8 — Upper bound on s

There is no (79, 109, 3350591)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 273 407251 594554 544872 455407 681690 621706 269036 453477 291703 438227 987557 578067 669745 265636 853960 048280 > 8¹⁰⁹ [i]