MinT - Best Known (88, 88+28, s)-Nets in Base 5

Best Known (88, 88+28, s)-Nets in Base 5

(88, 88+28, 408)-Net over F₅ — Constructive and digital

Digital (88, 116, 408)-net over F₅, using

trace code for nets [i] based on digital (30, 58, 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]

(88, 88+28, 3164)-Net over F₅ — Digital

Digital (88, 116, 3164)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹⁶, 3164, F₅, 28) (dual of [3164, 3048, 29]-code), using
- 24 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 6 times 0, 1, 14 times 0) [i] based on linear OA(5¹¹², 3136, F₅, 28) (dual of [3136, 3024, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. 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]
    2. linear OA(5¹⁰¹, 3125, F₅, 26) (dual of [3125, 3024, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [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, 88+28, 935101)-Net in Base 5 — Upper bound on s

There is no (88, 116, 935102)-net in base 5, because

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]