MinT - Best Known (17−6, 17, s)-Nets in Base 4

Best Known (17−6, 17, s)-Nets in Base 4

(17−6, 17, 86)-Net over F₄ — Constructive and digital

Digital (11, 17, 86)-net over F₄, using

net defined by OOA [i] based on linear OOA(4¹⁷, 86, F₄, 6, 6) (dual of [(86, 6), 499, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(4¹⁷, 258, F₄, 6) (dual of [258, 241, 7]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹⁷, 260, F₄, 6) (dual of [260, 243, 7]-code), using
    - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
      1. linear OA(4¹⁷, 256, F₄, 6) (dual of [256, 239, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      2. linear OA(4¹³, 256, F₄, 5) (dual of [256, 243, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
      3. linear OA(4⁰, 4, F₄, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(17−6, 17, 187)-Net over F₄ — Digital

Digital (11, 17, 187)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷, 187, F₄, 6) (dual of [187, 170, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷, 255, F₄, 6) (dual of [255, 238, 7]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(17−6, 17, 1560)-Net in Base 4 — Upper bound on s

There is no (11, 17, 1561)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 17182 587304 > 4¹⁷ [i]