MinT - Best Known (111−28, 111, s)-Nets in Base 9

Best Known (111−28, 111, s)-Nets in Base 9

(111−28, 111, 937)-Net over F₉ — Constructive and digital

Digital (83, 111, 937)-net over F₉, using

9¹ times duplication [i] based on digital (82, 110, 937)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹¹⁰, 937, F₉, 28, 28) (dual of [(937, 28), 26126, 29]-NRT-code), using
  - OA 14-folding and stacking [i] based on linear OA(9¹¹⁰, 13118, F₉, 28) (dual of [13118, 13008, 29]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹¹⁰, 13122, F₉, 28) (dual of [13122, 13012, 29]-code), using
      - trace code [i] based on linear OA(81⁵⁵, 6561, F₈₁, 28) (dual of [6561, 6506, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(111−28, 111, 13128)-Net over F₉ — Digital

Digital (83, 111, 13128)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹¹¹, 13128, F₉, 28) (dual of [13128, 13017, 29]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9¹¹⁰, 13126, F₉, 28) (dual of [13126, 13016, 29]-code), using
    - trace code [i] based on linear OA(81⁵⁵, 6563, F₈₁, 28) (dual of [6563, 6508, 29]-code), using
      - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
        linear OA(81⁵⁵, 6561, F₈₁, 28) (dual of [6561, 6506, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(81⁵³, 6561, F₈₁, 27) (dual of [6561, 6508, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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]
  2. linear OA(9¹¹⁰, 13127, F₉, 27) (dual of [13127, 13017, 28]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9¹¹⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 86 200676 142235 012215 619549 199577 703251 856613 280998 578887 011642 182925 923461 766876 807866 643992 501811 981809 [i]
  3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(111−28, 111, large)-Net in Base 9 — Upper bound on s

There is no (83, 111, large)-net in base 9, because

26 times m-reduction [i] would yield (83, 85, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111−28, 111, s)-Nets in Base 9

(111−28, 111, 937)-Net over F9 — Constructive and digital

(111−28, 111, 13128)-Net over F9 — Digital

(111−28, 111, large)-Net in Base 9 — Upper bound on s

(111−28, 111, 937)-Net over F₉ — Constructive and digital

(111−28, 111, 13128)-Net over F₉ — Digital