MinT - Best Known (91, 127, s)-Nets in Base 8

Best Known (91, 127, s)-Nets in Base 8

(91, 127, 514)-Net over F₈ — Constructive and digital

Digital (91, 127, 514)-net over F₈, using

8¹ times duplication [i] based on digital (90, 126, 514)-net over F₈, using
- t-expansion [i] based on digital (89, 126, 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 (51, 88, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 44, 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]

(91, 127, 600)-Net in Base 8 — Constructive

(91, 127, 600)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. (70, 106, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 53, 288)-net in base 64, using
    - 3 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]

(91, 127, 4111)-Net over F₈ — Digital

Digital (91, 127, 4111)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁷, 4111, F₈, 36) (dual of [4111, 3984, 37]-code), using
- 5 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 4 times 0) [i] based on linear OA(8¹²⁶, 4105, F₈, 36) (dual of [4105, 3979, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(33) [i] based on
    1. linear OA(8¹²⁵, 4096, F₈, 36) (dual of [4096, 3971, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    2. linear OA(8¹¹⁷, 4096, F₈, 34) (dual of [4096, 3979, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(91, 127, 2539988)-Net in Base 8 — Upper bound on s

There is no (91, 127, 2539989)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 925278 797374 491835 816303 372811 340083 403500 007471 558185 861808 176957 222701 914027 924791 426989 577203 496654 819255 866390 > 8¹²⁷ [i]