MinT - Best Known (238−21, 238, s)-Nets in Base 3

Best Known (238−21, 238, s)-Nets in Base 3

(238−21, 238, 838920)-Net over F₃ — Constructive and digital

Digital (217, 238, 838920)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (18, 28, 60)-net over F₃, using
  - trace code for nets [i] based on digital (4, 14, 30)-net over F₉, using
    - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
        a function field by SÃ©mirat [i]
2. digital (189, 210, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁰, 838860, F₃, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 8388601, F₃, 21) (dual of [8388601, 8388391, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(238−21, 238, 4194372)-Net over F₃ — Digital

Digital (217, 238, 4194372)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁸, 4194372, F₃, 2, 21) (dual of [(4194372, 2), 8388506, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁸, 71, F₃, 2, 10) (dual of [(71, 2), 114, 11]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁸, 71, F₃, 10) (dual of [71, 43, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁸, 92, F₃, 10) (dual of [92, 64, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(6) [i] based on
        linear OA(3²⁵, 81, F₃, 10) (dual of [81, 56, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(3¹⁷, 81, F₃, 7) (dual of [81, 64, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3³, 11, F₃, 2) (dual of [11, 8, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
  2. linear OOA(3²¹⁰, 4194301, F₃, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹⁰, 8388602, F₃, 21) (dual of [8388602, 8388392, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(238−21, 238, large)-Net in Base 3 — Upper bound on s

There is no (217, 238, large)-net in base 3, because

19 times m-reduction [i] would yield (217, 219, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (238−21, 238, s)-Nets in Base 3

(238−21, 238, 838920)-Net over F3 — Constructive and digital

(238−21, 238, 4194372)-Net over F3 — Digital

(238−21, 238, large)-Net in Base 3 — Upper bound on s

(238−21, 238, 838920)-Net over F₃ — Constructive and digital

(238−21, 238, 4194372)-Net over F₃ — Digital