MinT - Best Known (42−15, 42, s)-Nets in Base 128

Best Known (42−15, 42, s)-Nets in Base 128

(42−15, 42, 4681)-Net over F₁₂₈ — Constructive and digital

Digital (27, 42, 4681)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128⁴², 4681, F₁₂₈, 15, 15) (dual of [(4681, 15), 70173, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(128⁴², 32768, F₁₂₈, 15) (dual of [32768, 32726, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(128⁴², 32770, F₁₂₈, 15) (dual of [32770, 32728, 16]-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OA(128¹³, 16385, F₁₂₈, 7) (dual of [16385, 16372, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
      2. linear OA(128²⁹, 16385, F₁₂₈, 15) (dual of [16385, 16356, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(42−15, 42, 9619)-Net in Base 128 — Constructive

(27, 42, 9619)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (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. (19, 34, 9362)-net in base 128, using
  - net defined by OOA [i] based on OOA(128³⁴, 9362, S₁₂₈, 15, 15), using
    - OOA 7-folding and stacking with additional row [i] based on OA(128³⁴, 65535, S₁₂₈, 15), using
      - discarding factors based on OA(128³⁴, 65538, S₁₂₈, 15), using
        discarding parts of the base [i] based on linear OA(256²⁹, 65538, F₂₅₆, 15) (dual of [65538, 65509, 16]-code), using
        constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
        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(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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]

(42−15, 42, 99842)-Net over F₁₂₈ — Digital

Digital (27, 42, 99842)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−15, 42, large)-Net in Base 128 — Upper bound on s

There is no (27, 42, large)-net in base 128, because

13 times m-reduction [i] would yield (27, 29, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (42−15, 42, s)-Nets in Base 128

(42−15, 42, 4681)-Net over F128 — Constructive and digital

(42−15, 42, 9619)-Net in Base 128 — Constructive

(42−15, 42, 99842)-Net over F128 — Digital

(42−15, 42, large)-Net in Base 128 — Upper bound on s

(42−15, 42, 4681)-Net over F₁₂₈ — Constructive and digital

(42−15, 42, 99842)-Net over F₁₂₈ — Digital