MinT - Best Known (105−30, 105, s)-Nets in Base 25

Best Known (105−30, 105, s)-Nets in Base 25

(105−30, 105, 1108)-Net over F₂₅ — Constructive and digital

Digital (75, 105, 1108)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]
2. digital (56, 86, 1042)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁸⁶, 1042, F₂₅, 30, 30) (dual of [(1042, 30), 31174, 31]-NRT-code), using
    - OA 15-folding and stacking [i] based on linear OA(25⁸⁶, 15630, F₂₅, 30) (dual of [15630, 15544, 31]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁸⁶, 15632, F₂₅, 30) (dual of [15632, 15546, 31]-code), using
        constructionÂ X applied to C_e(29) âŠ‚ C_e(27) [i] based on
        linear OA(25⁸⁵, 15625, F₂₅, 30) (dual of [15625, 15540, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
        linear OA(25⁷⁹, 15625, F₂₅, 28) (dual of [15625, 15546, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(25¹, 7, F₂₅, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(105−30, 105, 56044)-Net over F₂₅ — Digital

Digital (75, 105, 56044)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−30, 105, large)-Net in Base 25 — Upper bound on s

There is no (75, 105, large)-net in base 25, because

28 times m-reduction [i] would yield (75, 77, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (105−30, 105, s)-Nets in Base 25

(105−30, 105, 1108)-Net over F25 — Constructive and digital

(105−30, 105, 56044)-Net over F25 — Digital

(105−30, 105, large)-Net in Base 25 — Upper bound on s

(105−30, 105, 1108)-Net over F₂₅ — Constructive and digital

(105−30, 105, 56044)-Net over F₂₅ — Digital