MinT - Best Known (98, 98+33, s)-Nets in Base 3

Best Known (98, 98+33, s)-Nets in Base 3

(98, 98+33, 282)-Net over F₃ — Constructive and digital

Digital (98, 131, 282)-net over F₃, using

1 times m-reduction [i] based on digital (98, 132, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 44, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(98, 98+33, 595)-Net over F₃ — Digital

Digital (98, 131, 595)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³¹, 595, F₃, 33) (dual of [595, 464, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³¹, 742, F₃, 33) (dual of [742, 611, 34]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(30) [i] based on
    1. linear OA(3¹³⁰, 729, F₃, 34) (dual of [729, 599, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹¹⁸, 729, F₃, 31) (dual of [729, 611, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(98, 98+33, 25577)-Net in Base 3 — Upper bound on s

There is no (98, 131, 25578)-net in base 3, because

1 times m-reduction [i] would yield (98, 130, 25578)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 106 153463 690902 333034 523315 778348 734556 234432 412167 532318 495649 > 3¹³⁰ [i]