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

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

(88−28, 88, 252)-Net over F₅ — Constructive and digital

Digital (60, 88, 252)-net over F₅, using

12 times m-reduction [i] based on digital (60, 100, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 50, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(88−28, 88, 559)-Net over F₅ — Digital

Digital (60, 88, 559)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁸, 559, F₅, 28) (dual of [559, 471, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁸, 632, F₅, 28) (dual of [632, 544, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. linear OA(5⁸⁷, 625, F₅, 28) (dual of [625, 538, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(5⁸¹, 625, F₅, 26) (dual of [625, 544, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(5¹, 7, F₅, 1) (dual of [7, 6, 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]

(88−28, 88, 37394)-Net in Base 5 — Upper bound on s

There is no (60, 88, 37395)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 32 318157 622780 583423 418277 135699 360633 995034 920687 130421 892553 > 5⁸⁸ [i]