MinT - Best Known (74, 103, s)-Nets in Base 9

Best Known (74, 103, s)-Nets in Base 9

(74, 103, 740)-Net over F₉ — Constructive and digital

Digital (74, 103, 740)-net over F₉, using

13 times m-reduction [i] based on digital (74, 116, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 58, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(74, 103, 5483)-Net over F₉ — Digital

Digital (74, 103, 5483)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁰³, 5483, F₉, 29) (dual of [5483, 5380, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹⁰³, 6571, F₉, 29) (dual of [6571, 6468, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(25) [i] based on
    1. linear OA(9¹⁰¹, 6561, F₉, 29) (dual of [6561, 6460, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(9⁹³, 6561, F₉, 26) (dual of [6561, 6468, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(9², 10, F₉, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
      - extended Reedâ€“Solomon code RS_e(8,9) [i]
      - Hamming code H(2,9) [i]

(74, 103, 6771821)-Net in Base 9 — Upper bound on s

There is no (74, 103, 6771822)-net in base 9, because

1 times m-reduction [i] would yield (74, 102, 6771822)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21 514752 227078 267135 297627 547388 794931 971391 281075 142831 212312 008995 658960 484894 674043 156889 319649 > 9¹⁰² [i]