MinT - Best Known (31−9, 31, s)-Nets in Base 27

Best Known (31−9, 31, s)-Nets in Base 27

(31−9, 31, 5272)-Net over F₂₇ — Constructive and digital

Digital (22, 31, 5272)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 351)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶, 351, F₂₇, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(27⁶, 702, F₂₇, 4) (dual of [702, 696, 5]-code), using
      - 1 times truncation [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (16, 25, 4921)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27²⁵, 4921, F₂₇, 9, 9) (dual of [(4921, 9), 44264, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(27²⁵, 19685, F₂₇, 9) (dual of [19685, 19660, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(27²⁵, 19686, F₂₇, 9) (dual of [19686, 19661, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(27²⁵, 19683, F₂₇, 9) (dual of [19683, 19658, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(27²², 19683, F₂₇, 8) (dual of [19683, 19661, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(27⁰, 3, F₂₇, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(31−9, 31, 50967)-Net over F₂₇ — Digital

Digital (22, 31, 50967)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−9, 31, large)-Net in Base 27 — Upper bound on s

There is no (22, 31, large)-net in base 27, because

7 times m-reduction [i] would yield (22, 24, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (31−9, 31, s)-Nets in Base 27

(31−9, 31, 5272)-Net over F27 — Constructive and digital

(31−9, 31, 50967)-Net over F27 — Digital

(31−9, 31, large)-Net in Base 27 — Upper bound on s

(31−9, 31, 5272)-Net over F₂₇ — Constructive and digital

(31−9, 31, 50967)-Net over F₂₇ — Digital