MinT - Best Known (259−49, 259, s)-Nets in Base 4

Best Known (259−49, 259, s)-Nets in Base 4

(259−49, 259, 1554)-Net over F₄ — Constructive and digital

Digital (210, 259, 1554)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (182, 231, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 77, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(259−49, 259, 12319)-Net over F₄ — Digital

Digital (210, 259, 12319)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁹, 12319, F₄, 49) (dual of [12319, 12060, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁹, 16411, F₄, 49) (dual of [16411, 16152, 50]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²⁵⁸, 16410, F₄, 49) (dual of [16410, 16152, 50]-code), using
    - constructionÂ X applied to C_e(48) âŠ‚ C_e(44) [i] based on
      1. linear OA(4²⁵³, 16384, F₄, 49) (dual of [16384, 16131, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
      2. linear OA(4²³², 16384, F₄, 45) (dual of [16384, 16152, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      3. linear OA(4⁵, 26, F₄, 3) (dual of [26, 21, 4]-code or 26-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]

(259−49, 259, large)-Net in Base 4 — Upper bound on s

There is no (210, 259, large)-net in base 4, because

47 times m-reduction [i] would yield (210, 212, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]