MinT - Best Known (23−9, 23, s)-Nets in Base 4

Best Known (23−9, 23, s)-Nets in Base 4

(23−9, 23, 66)-Net over F₄ — Constructive and digital

Digital (14, 23, 66)-net over F₄, using

1 times m-reduction [i] based on digital (14, 24, 66)-net over F₄, using
- trace code for nets [i] based on digital (2, 12, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]

(23−9, 23, 84)-Net over F₄ — Digital

Digital (14, 23, 84)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³, 84, F₄, 9) (dual of [84, 61, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4²³, 90, F₄, 9) (dual of [90, 67, 10]-code), using
  - constructionÂ X applied to C({2,5,6,14,17,21}) âŠ‚ C({2,5,6,14,17}) [i] based on
    1. linear OA(4²², 85, F₄, 9) (dual of [85, 63, 10]-code), using the cyclic code C(A) with length 85Â | 4⁴âˆ’1, defining set AÂ =Â {2,5,6,14,17,21}, and minimum distance dÂ â‰¥ |{−4,−1,2,…,20}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OA(4¹⁸, 85, F₄, 7) (dual of [85, 67, 8]-code), using the cyclic code C(A) with length 85Â | 4⁴âˆ’1, defining set AÂ =Â {2,5,6,14,17}, and minimum distance dÂ â‰¥ |{2,5,8,…,20}|+1Â =Â 8 (BCH-bound) [i]
    3. linear OA(4¹, 5, F₄, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(23−9, 23, 1508)-Net in Base 4 — Upper bound on s

There is no (14, 23, 1509)-net in base 4, because

1 times m-reduction [i] would yield (14, 22, 1509)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 17 631405 878329 > 4²² [i]