MinT - Best Known (151−23, 151, s)-Nets in Base 3

Best Known (151−23, 151, s)-Nets in Base 3

(151−23, 151, 5368)-Net over F₃ — Constructive and digital

Digital (128, 151, 5368)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁵¹, 5368, F₃, 23, 23) (dual of [(5368, 23), 123313, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using
  - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(151−23, 151, 19686)-Net over F₃ — Digital

Digital (128, 151, 19686)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵¹, 19686, F₃, 3, 23) (dual of [(19686, 3), 58907, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁵¹, 59058, F₃, 23) (dual of [59058, 58907, 24]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁵¹, 59059, F₃, 23) (dual of [59059, 58908, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
      1. linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(3¹⁴¹, 59049, F₃, 22) (dual of [59049, 58908, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(3⁰, 10, F₃, 0) (dual of [10, 10, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(151−23, 151, 7873766)-Net in Base 3 — Upper bound on s

There is no (128, 151, 7873767)-net in base 3, because

1 times m-reduction [i] would yield (128, 150, 7873767)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 369988 950440 764283 019421 702430 639616 236954 779733 659765 160588 860736 436235 > 3¹⁵⁰ [i]