MinT - Best Known (6, 6+9, s)-Nets in Base 128

Best Known (6, 6+9, s)-Nets in Base 128

(6, 6+9, 300)-Net over F₁₂₈ — Constructive and digital

Digital (6, 15, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (1, 10, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(6, 6+9, 385)-Net over F₁₂₈ — Digital

Digital (6, 15, 385)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128¹⁵, 385, F₁₂₈, 9) (dual of [385, 370, 10]-code), using
- constructionÂ XX applied to C₁Â =Â C([380,6]), C₂Â =Â C([0,7]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,6]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([380,7]) [i] based on
  1. linear OA(128¹³, 381, F₁₂₈, 8) (dual of [381, 368, 9]-code), using the BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â {−1,0,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  2. linear OA(128¹³, 381, F₁₂₈, 8) (dual of [381, 368, 9]-code), using the expurgated narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  3. linear OA(128¹⁵, 381, F₁₂₈, 9) (dual of [381, 366, 10]-code), using the BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â {−1,0,…,7}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
  4. linear OA(128¹¹, 381, F₁₂₈, 7) (dual of [381, 370, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
  5. linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]
  6. linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code) (see above)

(6, 6+9, 513)-Net in Base 128 — Constructive

(6, 15, 513)-net in base 128, using

net defined by OOA [i] based on OOA(128¹⁵, 513, S₁₂₈, 12, 9), using
- OOA stacking with additional row [i] based on OOA(128¹⁵, 514, S₁₂₈, 4, 9), using
  - discarding parts of the base [i] based on linear OOA(256¹³, 514, F₂₅₆, 4, 9) (dual of [(514, 4), 2043, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(256⁴, 257, F₂₅₆, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;1024,256) [i]
      2. linear OOA(256⁹, 257, F₂₅₆, 4, 9) (dual of [(257, 4), 1019, 10]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;1019,256) [i]

(6, 6+9, 413506)-Net in Base 128 — Upper bound on s

There is no (6, 15, 413507)-net in base 128, because

1 times m-reduction [i] would yield (6, 14, 413507)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 316914 083492 743730 147050 651152 > 128¹⁴ [i]

Best Known (6, 6+9, s)-Nets in Base 128

(6, 6+9, 300)-Net over F128 — Constructive and digital

(6, 6+9, 385)-Net over F128 — Digital

(6, 6+9, 513)-Net in Base 128 — Constructive

(6, 6+9, 413506)-Net in Base 128 — Upper bound on s

(6, 6+9, 300)-Net over F₁₂₈ — Constructive and digital

(6, 6+9, 385)-Net over F₁₂₈ — Digital