MinT - Best Known (127, 144, s)-Nets in Base 9

Best Known (127, 144, s)-Nets in Base 9

(127, 144, 2097184)-Net over F₉ — Constructive and digital

Digital (127, 144, 2097184)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 14, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (113, 130, 2097150)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹³⁰, 2097150, F₉, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(9¹³⁰, 8388601, F₉, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9¹³⁰, 8388602, F₉, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(81⁶⁵, 4194301, F₈₁, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁶⁵, 8388602, F₈₁, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁶⁵, large, F₈₁, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(127, 144, 2097188)-Net in Base 9 — Constructive

(127, 144, 2097188)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (6, 14, 38)-net in base 9, using
  - 1 times m-reduction [i] based on (6, 15, 38)-net in base 9, using
    - base change [i] based on digital (1, 10, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (113, 130, 2097150)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹³⁰, 2097150, F₉, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(9¹³⁰, 8388601, F₉, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9¹³⁰, 8388602, F₉, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(81⁶⁵, 4194301, F₈₁, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁶⁵, 8388602, F₈₁, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁶⁵, large, F₈₁, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(127, 144, large)-Net over F₉ — Digital

Digital (127, 144, large)-net over F₉, using

t-expansion [i] based on digital (124, 144, large)-net over F₉, using
- 1 times m-reduction [i] based on digital (124, 145, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(127, 144, large)-Net in Base 9 — Upper bound on s

There is no (127, 144, large)-net in base 9, because

15 times m-reduction [i] would yield (127, 129, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (127, 144, s)-Nets in Base 9

(127, 144, 2097184)-Net over F9 — Constructive and digital

(127, 144, 2097188)-Net in Base 9 — Constructive

(127, 144, large)-Net over F9 — Digital

(127, 144, large)-Net in Base 9 — Upper bound on s

(127, 144, 2097184)-Net over F₉ — Constructive and digital

(127, 144, large)-Net over F₉ — Digital