MinT - Best Known (26−5, 26, s)-Nets in Base 4

Best Known (26−5, 26, s)-Nets in Base 4

(26−5, 26, 65280)-Net over F₄ — Constructive and digital

Digital (21, 26, 65280)-net over F₄, using

net defined by OOA [i] based on linear OOA(4²⁶, 65280, F₄, 5, 5) (dual of [(65280, 5), 326374, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4²⁶, 130561, F₄, 5) (dual of [130561, 130535, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(4²⁶, 130562, F₄, 5) (dual of [130562, 130536, 6]-code), using
    - trace code [i] based on linear OA(16¹³, 65281, F₁₆, 5) (dual of [65281, 65268, 6]-code), using
      - base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]

(26−5, 26, 65281)-Net over F₄ — Digital

Digital (21, 26, 65281)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁶, 65281, F₄, 2, 5) (dual of [(65281, 2), 130536, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4²⁶, 130562, F₄, 5) (dual of [130562, 130536, 6]-code), using
  - trace code [i] based on linear OA(16¹³, 65281, F₁₆, 5) (dual of [65281, 65268, 6]-code), using
    - base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
      - a code by Danev and Olsson [i]

(26−5, 26, large)-Net in Base 4 — Upper bound on s

There is no (21, 26, large)-net in base 4, because

3 times m-reduction [i] would yield (21, 23, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]