MinT - Best Known (108−31, 108, s)-Nets in Base 7

Best Known (108−31, 108, s)-Nets in Base 7

(108−31, 108, 688)-Net over F₇ — Constructive and digital

Digital (77, 108, 688)-net over F₇, using

t-expansion [i] based on digital (76, 108, 688)-net over F₇, using
- 2 times m-reduction [i] based on digital (76, 110, 688)-net over F₇, using
  - trace code for nets [i] based on digital (21, 55, 344)-net over F₄₉, using
    - net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
        the Hermitian function field over F₄₉ [i]

(108−31, 108, 2431)-Net over F₇ — Digital

Digital (77, 108, 2431)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁰⁸, 2431, F₇, 31) (dual of [2431, 2323, 32]-code), using
- 23 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 17 times 0) [i] based on linear OA(7¹⁰⁵, 2405, F₇, 31) (dual of [2405, 2300, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(29) [i] based on
    1. linear OA(7¹⁰⁵, 2401, F₇, 31) (dual of [2401, 2296, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(7¹⁰¹, 2401, F₇, 30) (dual of [2401, 2300, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(108−31, 108, 1142819)-Net in Base 7 — Upper bound on s

There is no (77, 108, 1142820)-net in base 7, because

1 times m-reduction [i] would yield (77, 107, 1142820)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2 663733 544169 352394 132130 716726 164950 591991 399867 532660 854403 710321 397428 798617 910127 144993 > 7¹⁰⁷ [i]