MinT - Best Known (37−9, 37, s)-Nets in Base 128

Best Known (37−9, 37, s)-Nets in Base 128

(37−9, 37, 2097279)-Net over F₁₂₈ — Constructive and digital

Digital (28, 37, 2097279)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (24, 33, 2097150)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³³, 2097150, F₁₂₈, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(128³³, 8388601, F₁₂₈, 9) (dual of [8388601, 8388568, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(128³³, large, F₁₂₈, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 15790321Â | 128⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(37−9, 37, 2129919)-Net in Base 128 — Constructive

(28, 37, 2129919)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 8, 32769)-net in base 128, using
  - base change [i] based on digital (3, 7, 32769)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁷, 32769, F₂₅₆, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(256⁷, 65538, F₂₅₆, 4) (dual of [65538, 65531, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(256⁷, 65536, F₂₅₆, 4) (dual of [65536, 65529, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(256⁵, 65536, F₂₅₆, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]
2. (20, 29, 2097150)-net in base 128, using
  - net defined by OOA [i] based on OOA(128²⁹, 2097150, S₁₂₈, 9, 9), using
    - OOA 4-folding and stacking with additional row [i] based on OA(128²⁹, 8388601, S₁₂₈, 9), using
      - discarding factors based on OA(128²⁹, large, S₁₂₈, 9), using
        discarding parts of the base [i] based on linear OA(256²⁵, large, F₂₅₆, 9) (dual of [large, large−25, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(37−9, 37, large)-Net over F₁₂₈ — Digital

Digital (28, 37, large)-net over F₁₂₈, using

128¹ times duplication [i] based on digital (27, 36, large)-net over F₁₂₈, using
- t-expansion [i] based on digital (26, 36, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(37−9, 37, large)-Net in Base 128 — Upper bound on s

There is no (28, 37, large)-net in base 128, because

7 times m-reduction [i] would yield (28, 30, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (37−9, 37, s)-Nets in Base 128

(37−9, 37, 2097279)-Net over F128 — Constructive and digital

(37−9, 37, 2129919)-Net in Base 128 — Constructive

(37−9, 37, large)-Net over F128 — Digital

(37−9, 37, large)-Net in Base 128 — Upper bound on s

(37−9, 37, 2097279)-Net over F₁₂₈ — Constructive and digital

(37−9, 37, large)-Net over F₁₂₈ — Digital