MinT - Best Known (248−46, 248, s)-Nets in Base 4

Best Known (248−46, 248, s)-Nets in Base 4

(248−46, 248, 1553)-Net over F₄ — Constructive and digital

Digital (202, 248, 1553)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 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 (176, 222, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 74, 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]

(248−46, 248, 13754)-Net over F₄ — Digital

Digital (202, 248, 13754)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁸, 13754, F₄, 46) (dual of [13754, 13506, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁴⁸, 16421, F₄, 46) (dual of [16421, 16173, 47]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(4²⁴⁶, 16419, F₄, 46) (dual of [16419, 16173, 47]-code), using
    - constructionÂ X applied to C_e(45) âŠ‚ C_e(40) [i] based on
      1. linear OA(4²³⁹, 16384, F₄, 46) (dual of [16384, 16145, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
      2. linear OA(4²¹¹, 16384, F₄, 41) (dual of [16384, 16173, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(248−46, 248, large)-Net in Base 4 — Upper bound on s

There is no (202, 248, large)-net in base 4, because

44 times m-reduction [i] would yield (202, 204, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]