MinT - Best Known (21−5, 21, s)-Nets in Base 4

Best Known (21−5, 21, s)-Nets in Base 4

(21−5, 21, 6049)-Net over F₄ — Constructive and digital

Digital (16, 21, 6049)-net over F₄, using

net defined by OOA [i] based on linear OOA(4²¹, 6049, F₄, 5, 5) (dual of [(6049, 5), 30224, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4²¹, 12099, F₄, 5) (dual of [12099, 12078, 6]-code), using
  - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
    - a code by Danev and Olsson [i]

(21−5, 21, 6250)-Net over F₄ — Digital

Digital (16, 21, 6250)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹, 6250, F₄, 5) (dual of [6250, 6229, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹, 12099, F₄, 5) (dual of [12099, 12078, 6]-code), using
  - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
    - a code by Danev and Olsson [i]

(21−5, 21, 494302)-Net in Base 4 — Upper bound on s

There is no (16, 21, 494303)-net in base 4, because

1 times m-reduction [i] would yield (16, 20, 494303)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 099514 741323 > 4²⁰ [i]