MinT - Best Known (36−13, 36, s)-Nets in Base 128

Best Known (36−13, 36, s)-Nets in Base 128

(36−13, 36, 5461)-Net over F₁₂₈ — Constructive and digital

Digital (23, 36, 5461)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128³⁶, 5461, F₁₂₈, 13, 13) (dual of [(5461, 13), 70957, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(128³⁶, 32767, F₁₂₈, 13) (dual of [32767, 32731, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(128³⁶, 32769, F₁₂₈, 13) (dual of [32769, 32733, 14]-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OA(128¹¹, 16384, F₁₂₈, 6) (dual of [16384, 16373, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      2. linear OA(128²⁵, 16385, F₁₂₈, 13) (dual of [16385, 16360, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(36−13, 36, 11179)-Net in Base 128 — Constructive

(23, 36, 11179)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (1, 7, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
    - base change [i] based on digital (0, 7, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. (16, 29, 10922)-net in base 128, using
  - net defined by OOA [i] based on OOA(128²⁹, 10922, S₁₂₈, 13, 13), using
    - OOA 6-folding and stacking with additional row [i] based on OA(128²⁹, 65533, S₁₂₈, 13), using
      - discarding factors based on OA(128²⁹, 65538, S₁₂₈, 13), using
        discarding parts of the base [i] based on linear OA(256²⁵, 65538, F₂₅₆, 13) (dual of [65538, 65513, 14]-code), using
        constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(256²⁵, 65536, F₂₅₆, 13) (dual of [65536, 65511, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [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]

(36−13, 36, 87341)-Net over F₁₂₈ — Digital

Digital (23, 36, 87341)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(36−13, 36, large)-Net in Base 128 — Upper bound on s

There is no (23, 36, large)-net in base 128, because

11 times m-reduction [i] would yield (23, 25, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (36−13, 36, s)-Nets in Base 128

(36−13, 36, 5461)-Net over F128 — Constructive and digital

(36−13, 36, 11179)-Net in Base 128 — Constructive

(36−13, 36, 87341)-Net over F128 — Digital

(36−13, 36, large)-Net in Base 128 — Upper bound on s

(36−13, 36, 5461)-Net over F₁₂₈ — Constructive and digital

(36−13, 36, 87341)-Net over F₁₂₈ — Digital