MinT - Best Known (58, 58+13, s)-Nets in Base 4

Best Known (58, 58+13, s)-Nets in Base 4

(58, 58+13, 2739)-Net over F₄ — Constructive and digital

Digital (58, 71, 2739)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (51, 64, 2730)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁶⁴, 2730, F₄, 13, 13) (dual of [(2730, 13), 35426, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(4⁶⁴, 16381, F₄, 13) (dual of [16381, 16317, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁶⁴, 16384, F₄, 13) (dual of [16384, 16320, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(58, 58+13, 11088)-Net over F₄ — Digital

Digital (58, 71, 11088)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷¹, 11088, F₄, 13) (dual of [11088, 11017, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷¹, 16385, F₄, 13) (dual of [16385, 16314, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(58, 58+13, large)-Net in Base 4 — Upper bound on s

There is no (58, 71, large)-net in base 4, because

11 times m-reduction [i] would yield (58, 60, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]