MinT - Best Known (16−5, 16, s)-Nets in Base 7

Best Known (16−5, 16, s)-Nets in Base 7

(16−5, 16, 2360)-Net over F₇ — Constructive and digital

Digital (11, 16, 2360)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹⁶, 2360, F₇, 6, 5) (dual of [(2360, 6), 14144, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7¹⁶, 2361, F₇, 2, 5) (dual of [(2361, 2), 4706, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(7², 8, F₇, 2, 2) (dual of [(8, 2), 14, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(2;14,7) [i]
    2. linear OOA(7¹⁴, 2353, F₇, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(7¹⁴, 4706, F₇, 5) (dual of [4706, 4692, 6]-code), using
        trace code [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(16−5, 16, 4714)-Net over F₇ — Digital

Digital (11, 16, 4714)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁶, 4714, F₇, 5) (dual of [4714, 4698, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(7², 8, F₇, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
    - extended Reedâ€“Solomon code RS_e(6,7) [i]
    - Hamming code H(2,7) [i]
    - algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using the rational function field F₇(x) [i]
  2. linear OA(7¹⁴, 4706, F₇, 5) (dual of [4706, 4692, 6]-code), using
    - trace code [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
      - a code by Danev and Olsson [i]

(16−5, 16, 513568)-Net in Base 7 — Upper bound on s

There is no (11, 16, 513569)-net in base 7, because

1 times m-reduction [i] would yield (11, 15, 513569)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4 747571 526769 > 7¹⁵ [i]