MinT - Best Known (30, 39, s)-Nets in Base 9

Best Known (30, 39, s)-Nets in Base 9

(30, 39, 3297)-Net over F₉ — Constructive and digital

Digital (30, 39, 3297)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (25, 34, 3281)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9³⁴, 3281, F₉, 9, 9) (dual of [(3281, 9), 29495, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(9³⁴, 13125, F₉, 9) (dual of [13125, 13091, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(9³⁴, 13126, F₉, 9) (dual of [13126, 13092, 10]-code), using
        trace code [i] based on linear OA(81¹⁷, 6563, F₈₁, 9) (dual of [6563, 6546, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(81¹⁷, 6561, F₈₁, 9) (dual of [6561, 6544, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(30, 39, 4922)-Net in Base 9 — Constructive

(30, 39, 4922)-net in base 9, using

base change [i] based on digital (17, 26, 4922)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27²⁶, 4922, F₂₇, 9, 9) (dual of [(4922, 9), 44272, 10]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OA(27²⁶, 19689, F₂₇, 9) (dual of [19689, 19663, 10]-code), using
    - discarding factors / shortening the dual code based on linear OA(27²⁶, 19691, F₂₇, 9) (dual of [19691, 19665, 10]-code), using
      - constructionÂ X applied to C([0,4]) âŠ‚ C([0,3]) [i] based on
        linear OA(27²⁵, 19684, F₂₇, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(27¹⁹, 19684, F₂₇, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(27¹, 7, F₂₇, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(30, 39, 21116)-Net over F₉ — Digital

Digital (30, 39, 21116)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 39, large)-Net in Base 9 — Upper bound on s

There is no (30, 39, large)-net in base 9, because

7 times m-reduction [i] would yield (30, 32, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (30, 39, s)-Nets in Base 9

(30, 39, 3297)-Net over F9 — Constructive and digital

(30, 39, 4922)-Net in Base 9 — Constructive

(30, 39, 21116)-Net over F9 — Digital

(30, 39, large)-Net in Base 9 — Upper bound on s

(30, 39, 3297)-Net over F₉ — Constructive and digital

(30, 39, 21116)-Net over F₉ — Digital