MinT - Best Known (129−55, 129, s)-Nets in Base 16

Best Known (129−55, 129, s)-Nets in Base 16

(129−55, 129, 532)-Net over F₁₆ — Constructive and digital

Digital (74, 129, 532)-net over F₁₆, using

1 times m-reduction [i] based on digital (74, 130, 532)-net over F₁₆, using
- trace code for nets [i] based on digital (9, 65, 266)-net over F₂₅₆, using
  - net from sequence [i] based on digital (9, 265)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(129−55, 129, 1080)-Net over F₁₆ — Digital

Digital (74, 129, 1080)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁹, 1080, F₁₆, 55) (dual of [1080, 951, 56]-code), using
- 50 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 10 times 0, 1, 38 times 0) [i] based on linear OA(16¹²⁷, 1028, F₁₆, 55) (dual of [1028, 901, 56]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(16¹²⁶, 1026, F₁₆, 55) (dual of [1026, 900, 56]-code), using
      - trace code [i] based on linear OA(256⁶³, 513, F₂₅₆, 55) (dual of [513, 450, 56]-code), using
        extended algebraic-geometric code AG_e(F,457P) [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]
    2. linear OA(16¹²⁶, 1027, F₁₆, 54) (dual of [1027, 901, 55]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹²⁶Â > V_b^sâˆ’1(kâˆ’1)Â = 50 210790 698738 752486 317399 364186 941159 934807 596008 383676 989017 812331 765289 235222 248624 068592 070976 287060 287791 823211 609399 347728 684299 524596 332472 726016 [i]
    3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(129−55, 129, 372150)-Net in Base 16 — Upper bound on s

There is no (74, 129, 372151)-net in base 16, because

1 times m-reduction [i] would yield (74, 128, 372151)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13408 051058 584558 809075 036531 052863 028706 448768 947556 968988 172082 369701 596369 880319 562566 302730 785620 426726 113840 851454 923875 555927 860082 248501 588792 538256 > 16¹²⁸ [i]