MinT - Best Known (82−19, 82, s)-Nets in Base 25

Best Known (82−19, 82, s)-Nets in Base 25

(82−19, 82, 43429)-Net over F₂₅ — Constructive and digital

Digital (63, 82, 43429)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
      - the rational function field F₂₅(x) [i]
    - Niederreiter sequence [i]
2. digital (54, 73, 43403)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷³, 43403, F₂₅, 19, 19) (dual of [(43403, 19), 824584, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(25⁷³, 390628, F₂₅, 19) (dual of [390628, 390555, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁷³, 390629, F₂₅, 19) (dual of [390629, 390556, 20]-code), using
        constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
        linear OA(25⁷³, 390625, F₂₅, 19) (dual of [390625, 390552, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(25⁶⁹, 390625, F₂₅, 18) (dual of [390625, 390556, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(25⁰, 4, F₂₅, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(82−19, 82, 735051)-Net over F₂₅ — Digital

Digital (63, 82, 735051)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−19, 82, large)-Net in Base 25 — Upper bound on s

There is no (63, 82, large)-net in base 25, because

17 times m-reduction [i] would yield (63, 65, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (82−19, 82, s)-Nets in Base 25

(82−19, 82, 43429)-Net over F25 — Constructive and digital

(82−19, 82, 735051)-Net over F25 — Digital

(82−19, 82, large)-Net in Base 25 — Upper bound on s

(82−19, 82, 43429)-Net over F₂₅ — Constructive and digital

(82−19, 82, 735051)-Net over F₂₅ — Digital