MinT - Best Known (16−5, 16, s)-Nets in Base 5

Best Known (16−5, 16, s)-Nets in Base 5

(16−5, 16, 606)-Net over F₅ — Constructive and digital

Digital (11, 16, 606)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁶, 606, F₅, 6, 5) (dual of [(606, 6), 3620, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5¹⁶, 607, F₅, 2, 5) (dual of [(607, 2), 1198, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5², 6, F₅, 2, 2) (dual of [(6, 2), 10, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(2;10,5) [i]
    2. linear OOA(5¹⁴, 601, F₅, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
        trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]

(16−5, 16, 1208)-Net over F₅ — Digital

Digital (11, 16, 1208)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁶, 1208, F₅, 5) (dual of [1208, 1192, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(5², 6, F₅, 2) (dual of [6, 4, 3]-code or 6-arc in PG(1,5)), using
    - extended Reedâ€“Solomon code RS_e(4,5) [i]
    - Hamming code H(2,5) [i]
  2. linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
    - trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
      - a code by Danev and Olsson [i]

(16−5, 16, 61762)-Net in Base 5 — Upper bound on s

There is no (11, 16, 61763)-net in base 5, because

1 times m-reduction [i] would yield (11, 15, 61763)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 30518 333561 > 5¹⁵ [i]