MinT - Best Known (121−29, 121, s)-Nets in Base 5

Best Known (121−29, 121, s)-Nets in Base 5

(121−29, 121, 408)-Net over F₅ — Constructive and digital

Digital (92, 121, 408)-net over F₅, using

3 times m-reduction [i] based on digital (92, 124, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 62, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(121−29, 121, 3199)-Net over F₅ — Digital

Digital (92, 121, 3199)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²¹, 3199, F₅, 29) (dual of [3199, 3078, 30]-code), using
- 64 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 36 times 0) [i] based on linear OA(5¹¹⁶, 3130, F₅, 29) (dual of [3130, 3014, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(5¹¹⁶, 3125, F₅, 29) (dual of [3125, 3009, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(5¹¹¹, 3125, F₅, 28) (dual of [3125, 3014, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(121−29, 121, 1481038)-Net in Base 5 — Upper bound on s

There is no (92, 121, 1481039)-net in base 5, because

1 times m-reduction [i] would yield (92, 120, 1481039)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 752321 559721 823197 030697 042519 435401 308948 901864 283010 533729 315024 581624 116073 218505 > 5¹²⁰ [i]