MinT - Best Known (97−21, 97, s)-Nets in Base 4

Best Known (97−21, 97, s)-Nets in Base 4

(97−21, 97, 1042)-Net over F₄ — Constructive and digital

Digital (76, 97, 1042)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (63, 84, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 21, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(97−21, 97, 2897)-Net over F₄ — Digital

Digital (76, 97, 2897)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁷, 2897, F₄, 21) (dual of [2897, 2800, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹⁷, 4097, F₄, 21) (dual of [4097, 4000, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(97−21, 97, 909132)-Net in Base 4 — Upper bound on s

There is no (76, 97, 909133)-net in base 4, because

1 times m-reduction [i] would yield (76, 96, 909133)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6277 130478 249313 580515 722400 288312 341544 852042 829611 076530 > 4⁹⁶ [i]