MinT - Best Known (72, 98, s)-Nets in Base 8

Best Known (72, 98, s)-Nets in Base 8

(72, 98, 562)-Net over F₈ — Constructive and digital

Digital (72, 98, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (19, 32, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 16, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (40, 66, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 33, 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, 98, 674)-Net in Base 8 — Constructive

(72, 98, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (15, 28, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 14, 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. (44, 70, 514)-net in base 8, using
  - trace code for nets [i] based on (9, 35, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
      - base change [i] based on digital (0, 27, 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, 98, 5062)-Net over F₈ — Digital

Digital (72, 98, 5062)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁸, 5062, F₈, 26) (dual of [5062, 4964, 27]-code), using
- 953 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 10 times 0, 1, 29 times 0, 1, 76 times 0, 1, 168 times 0, 1, 285 times 0, 1, 375 times 0) [i] based on linear OA(8⁸⁹, 4100, F₈, 26) (dual of [4100, 4011, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(24) [i] based on
    1. linear OA(8⁸⁹, 4096, F₈, 26) (dual of [4096, 4007, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(8⁸⁵, 4096, F₈, 25) (dual of [4096, 4011, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(72, 98, 5202554)-Net in Base 8 — Upper bound on s

There is no (72, 98, 5202555)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31828 749215 529247 297177 987065 222236 642739 645832 293693 101277 070994 732976 824786 069227 583064 > 8⁹⁸ [i]