MinT - Best Known (103, 117, s)-Nets in Base 3

Best Known (103, 117, s)-Nets in Base 3

(103, 117, 75928)-Net over F₃ — Constructive and digital

Digital (103, 117, 75928)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 109, 75921)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁹, 75921, F₃, 14, 14) (dual of [(75921, 14), 1062785, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(3¹⁰⁹, 531447, F₃, 14) (dual of [531447, 531338, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁹, 531453, F₃, 14) (dual of [531453, 531344, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(3¹⁰⁹, 531441, F₃, 14) (dual of [531441, 531332, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(3⁹⁷, 531441, F₃, 13) (dual of [531441, 531344, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁰, 12, F₃, 0) (dual of [12, 12, 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]

(103, 117, 238810)-Net over F₃ — Digital

Digital (103, 117, 238810)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁷, 238810, F₃, 2, 14) (dual of [(238810, 2), 477503, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹¹⁷, 265742, F₃, 2, 14) (dual of [(265742, 2), 531367, 15]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹¹⁶, 265742, F₃, 2, 14) (dual of [(265742, 2), 531368, 15]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹¹⁶, 531484, F₃, 14) (dual of [531484, 531368, 15]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹¹⁵, 531483, F₃, 14) (dual of [531483, 531368, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(9) [i] based on
        linear OA(3¹⁰⁹, 531441, F₃, 14) (dual of [531441, 531332, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(3⁷³, 531441, F₃, 10) (dual of [531441, 531368, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(3⁶, 42, F₃, 3) (dual of [42, 36, 4]-code or 42-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(103, 117, large)-Net in Base 3 — Upper bound on s

There is no (103, 117, large)-net in base 3, because

12 times m-reduction [i] would yield (103, 105, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (103, 117, s)-Nets in Base 3

(103, 117, 75928)-Net over F3 — Constructive and digital

(103, 117, 238810)-Net over F3 — Digital

(103, 117, large)-Net in Base 3 — Upper bound on s

(103, 117, 75928)-Net over F₃ — Constructive and digital

(103, 117, 238810)-Net over F₃ — Digital