MinT - Best Known (59, 78, s)-Nets in Base 7

Best Known (59, 78, s)-Nets in Base 7

(59, 78, 535)-Net over F₇ — Constructive and digital

Digital (59, 78, 535)-net over F₇, using

net defined by OOA [i] based on linear OOA(7⁷⁸, 535, F₇, 19, 19) (dual of [(535, 19), 10087, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(7⁷⁸, 4816, F₇, 19) (dual of [4816, 4738, 20]-code), using
  - discarding factors / shortening the dual code based on linear OA(7⁷⁸, 4818, F₇, 19) (dual of [4818, 4740, 20]-code), using
    - trace code [i] based on linear OA(49³⁹, 2409, F₄₉, 19) (dual of [2409, 2370, 20]-code), using
      - constructionÂ X applied to C_e(18) âŠ‚ C_e(15) [i] based on
        linear OA(49³⁷, 2401, F₄₉, 19) (dual of [2401, 2364, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(49³¹, 2401, F₄₉, 16) (dual of [2401, 2370, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(49², 8, F₄₉, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
        discarding factors / shortening the dual code based on linear OA(49², 49, F₄₉, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
        Reedâ€“Solomon code RS(47,49) [i]

(59, 78, 5791)-Net over F₇ — Digital

Digital (59, 78, 5791)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 78, large)-Net in Base 7 — Upper bound on s

There is no (59, 78, large)-net in base 7, because

17 times m-reduction [i] would yield (59, 61, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]