MinT - Best Known (45−16, 45, s)-Nets in Base 128

Best Known (45−16, 45, s)-Nets in Base 128

(45−16, 45, 2435)-Net over F₁₂₈ — Constructive and digital

Digital (29, 45, 2435)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (6, 14, 387)-net over F₁₂₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 129)-net over F₁₂₈, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 4, 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]
    3. digital (0, 8, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (15, 31, 2048)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³¹, 2048, F₁₂₈, 16, 16) (dual of [(2048, 16), 32737, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(128³¹, 16384, F₁₂₈, 16) (dual of [16384, 16353, 17]-code), using
      - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(45−16, 45, 8342)-Net in Base 128 — Constructive

(29, 45, 8342)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. (20, 36, 8192)-net in base 128, using
  - net defined by OOA [i] based on OOA(128³⁶, 8192, S₁₂₈, 16, 16), using
    - OA 8-folding and stacking [i] based on OA(128³⁶, 65536, S₁₂₈, 16), using
      - discarding factors based on OA(128³⁶, 65538, S₁₂₈, 16), using
        discarding parts of the base [i] based on linear OA(256³¹, 65538, F₂₅₆, 16) (dual of [65538, 65507, 17]-code), using
        constructionÂ X applied to C_e(15) âŠ‚ C_e(14) [i] based on
        linear OA(256³¹, 65536, F₂₅₆, 16) (dual of [65536, 65505, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(256²⁹, 65536, F₂₅₆, 15) (dual of [65536, 65507, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(45−16, 45, 106078)-Net over F₁₂₈ — Digital

Digital (29, 45, 106078)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(45−16, 45, large)-Net in Base 128 — Upper bound on s

There is no (29, 45, large)-net in base 128, because

14 times m-reduction [i] would yield (29, 31, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (45−16, 45, s)-Nets in Base 128

(45−16, 45, 2435)-Net over F128 — Constructive and digital

(45−16, 45, 8342)-Net in Base 128 — Constructive

(45−16, 45, 106078)-Net over F128 — Digital

(45−16, 45, large)-Net in Base 128 — Upper bound on s

(45−16, 45, 2435)-Net over F₁₂₈ — Constructive and digital

(45−16, 45, 106078)-Net over F₁₂₈ — Digital