MinT - Best Known (31−9, 31, s)-Nets in Base 49

Best Known (31−9, 31, s)-Nets in Base 49

(31−9, 31, 30588)-Net over F₄₉ — Constructive and digital

Digital (22, 31, 30588)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 1176)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49⁶, 1176, F₄₉, 4, 4) (dual of [(1176, 4), 4698, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(49⁶, 1176, F₄₉, 3, 4) (dual of [(1176, 3), 3522, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(49⁶, 2352, F₄₉, 4) (dual of [2352, 2346, 5]-code), using
        1 times truncation [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (16, 25, 29412)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49²⁵, 29412, F₄₉, 9, 9) (dual of [(29412, 9), 264683, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(49²⁵, 117649, F₄₉, 9) (dual of [117649, 117624, 10]-code), using
      - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(31−9, 31, 277949)-Net over F₄₉ — Digital

Digital (22, 31, 277949)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−9, 31, large)-Net in Base 49 — Upper bound on s

There is no (22, 31, large)-net in base 49, because

7 times m-reduction [i] would yield (22, 24, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]