MinT - Best Known (13−6, 13, s)-Nets in Base 4

Best Known (13−6, 13, s)-Nets in Base 4

(13−6, 13, 35)-Net over F₄ — Constructive and digital

Digital (7, 13, 35)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 4, 17)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
2. digital (3, 9, 18)-net over F₄, using
  - linear code embedding found in a computer search [i]

(13−6, 13, 45)-Net over F₄ — Digital

Digital (7, 13, 45)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³, 45, F₄, 6) (dual of [45, 32, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³, 63, F₄, 6) (dual of [63, 50, 7]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(13−6, 13, 244)-Net in Base 4 — Upper bound on s

There is no (7, 13, 245)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 67 805956 > 4¹³ [i]