MinT - Best Known (88, 117, s)-Nets in Base 5

Best Known (88, 117, s)-Nets in Base 5

(88, 117, 400)-Net over F₅ — Constructive and digital

Digital (88, 117, 400)-net over F₅, using

9 times m-reduction [i] based on digital (88, 126, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 63, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

(88, 117, 2731)-Net over F₅ — Digital

Digital (88, 117, 2731)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹⁷, 2731, F₅, 29) (dual of [2731, 2614, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹¹⁷, 3136, F₅, 29) (dual of [3136, 3019, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(26) [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(5¹⁰⁶, 3125, F₅, 27) (dual of [3125, 3019, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(88, 117, 935101)-Net in Base 5 — Upper bound on s

There is no (88, 117, 935102)-net in base 5, because

1 times m-reduction [i] would yield (88, 116, 935102)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1203 711331 852872 677345 547766 898283 082923 882665 167969 578019 630311 969533 809329 199425 > 5¹¹⁶ [i]