MinT - Best Known (37−22, 37, s)-Nets in Base 128

Best Known (37−22, 37, s)-Nets in Base 128

(37−22, 37, 342)-Net over F₁₂₈ — Constructive and digital

Digital (15, 37, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 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. digital (3, 25, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(37−22, 37, 386)-Net in Base 128 — Constructive

(15, 37, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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. (4, 26, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
    - base change [i] based on digital (0, 28, 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]

(37−22, 37, 388)-Net over F₁₂₈ — Digital

Digital (15, 37, 388)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128³⁷, 388, F₁₂₈, 22) (dual of [388, 351, 23]-code), using
- 4 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 0) [i] based on linear OA(128³⁶, 383, F₁₂₈, 22) (dual of [383, 347, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. linear OA(128³⁶, 382, F₁₂₈, 22) (dual of [382, 346, 23]-code), using an extension C_e(21) of the narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(128³⁵, 382, F₁₂₈, 21) (dual of [382, 347, 22]-code), using an extension C_e(20) of the narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(128⁰, 1, F₁₂₈, 0) (dual of [1, 1, 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]

(37−22, 37, 513)-Net in Base 128

(15, 37, 513)-net in base 128, using

19 times m-reduction [i] based on (15, 56, 513)-net in base 128, using
- base change [i] based on digital (8, 49, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(37−22, 37, 473253)-Net in Base 128 — Upper bound on s

There is no (15, 37, 473254)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 926347 183985 489159 557962 946278 744025 216742 748551 958003 405276 993892 865531 090056 > 128³⁷ [i]