MinT - Best Known (25, 36, s)-Nets in Base 128

Best Known (25, 36, s)-Nets in Base 128

(25, 36, 419559)-Net over F₁₂₈ — Constructive and digital

Digital (25, 36, 419559)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (20, 31, 419430)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³¹, 419430, F₁₂₈, 11, 11) (dual of [(419430, 11), 4613699, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(128³¹, 2097151, F₁₂₈, 11) (dual of [2097151, 2097120, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(128³¹, 2097152, F₁₂₈, 11) (dual of [2097152, 2097121, 12]-code), using
        an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(25, 36, 1677720)-Net in Base 128 — Constructive

(25, 36, 1677720)-net in base 128, using

net defined by OOA [i] based on OOA(128³⁶, 1677720, S₁₂₈, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(128³⁶, 8388601, S₁₂₈, 11), using
  - discarding factors based on OA(128³⁶, large, S₁₂₈, 11), using
    - discarding parts of the base [i] based on linear OA(256³¹, large, F₂₅₆, 11) (dual of [large, large−31, 12]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(25, 36, 2097284)-Net over F₁₂₈ — Digital

Digital (25, 36, 2097284)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128³⁶, 2097284, F₁₂₈, 11) (dual of [2097284, 2097248, 12]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(128⁵, 129, F₁₂₈, 5) (dual of [129, 124, 6]-code or 129-arc in PG(4,128)), using
    - extended Reedâ€“Solomon code RS_e(124,128) [i]
  2. linear OA(128³¹, 2097155, F₁₂₈, 11) (dual of [2097155, 2097124, 12]-code), using
    - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
      1. linear OA(128³¹, 2097152, F₁₂₈, 11) (dual of [2097152, 2097121, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      2. linear OA(128²⁸, 2097152, F₁₂₈, 10) (dual of [2097152, 2097124, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(25, 36, large)-Net in Base 128 — Upper bound on s

There is no (25, 36, large)-net in base 128, because

9 times m-reduction [i] would yield (25, 27, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]