MinT - Best Known (59−6, 59, s)-Nets in Base 4

Best Known (59−6, 59, s)-Nets in Base 4

(59−6, 59, 2801241)-Net over F₄ — Constructive and digital

Digital (53, 59, 2801241)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 10, 5040)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁰, 5040, F₄, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(4¹⁰, 5040, F₄, 2, 3) (dual of [(5040, 2), 10070, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4¹⁰, 5040, F₄, 3) (dual of [5040, 5030, 4]-code or 5040-cap in PG(9,4)), using
        product of projective cap with cap in AG(4,Â 4) [i] based on linear OA(4⁶, 126, F₄, 3) (dual of [126, 120, 4]-code or 126-cap in PG(5,4)), using
        the Glynn cap [i]
2. digital (43, 49, 2796201)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴⁹, 2796201, F₄, 6, 6) (dual of [(2796201, 6), 16777157, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(4⁴⁹, large, F₄, 6) (dual of [large, large−49, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(59−6, 59, large)-Net over F₄ — Digital

Digital (53, 59, large)-net over F₄, using

4¹⁰ times duplication [i] based on digital (43, 49, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁹, large, F₄, 6) (dual of [large, large−49, 7]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(59−6, 59, large)-Net in Base 4 — Upper bound on s

There is no (53, 59, large)-net in base 4, because

4 times m-reduction [i] would yield (53, 55, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]