MinT - Best Known (78−17, 78, s)-Nets in Base 4

Best Known (78−17, 78, s)-Nets in Base 4

(78−17, 78, 1038)-Net over F₄ — Constructive and digital

Digital (61, 78, 1038)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  - a shift-net [i]
2. digital (51, 68, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 17, 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]

(78−17, 78, 2626)-Net over F₄ — Digital

Digital (61, 78, 2626)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁸, 2626, F₄, 17) (dual of [2626, 2548, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁸, 4119, F₄, 17) (dual of [4119, 4041, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
    1. linear OA(4⁷³, 4096, F₄, 17) (dual of [4096, 4023, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(4⁵⁵, 4096, F₄, 13) (dual of [4096, 4041, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(4⁵, 23, F₄, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(78−17, 78, 782335)-Net in Base 4 — Upper bound on s

There is no (61, 78, 782336)-net in base 4, because

1 times m-reduction [i] would yield (61, 77, 782336)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 22836 178131 941162 914563 956137 748231 261633 324545 > 4⁷⁷ [i]