MinT - Best Known (162−28, 162, s)-Nets in Base 4

Best Known (162−28, 162, s)-Nets in Base 4

(162−28, 162, 1272)-Net over F₄ — Constructive and digital

Digital (134, 162, 1272)-net over F₄, using

4¹ times duplication [i] based on digital (133, 161, 1272)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (31, 45, 240)-net over F₄, using
    - trace code for nets [i] based on digital (1, 15, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (88, 116, 1032)-net over F₄, using
    - trace code for nets [i] based on digital (1, 29, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(162−28, 162, 1286)-Net in Base 4 — Constructive

(134, 162, 1286)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (36, 50, 258)-net in base 4, using
  - trace code for nets [i] based on (11, 25, 129)-net in base 16, using
    - base change [i] based on (6, 20, 129)-net in base 32, using
      - 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
        base change [i] based on digital (0, 15, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. digital (84, 112, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 28, 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]

(162−28, 162, 16441)-Net over F₄ — Digital

Digital (134, 162, 16441)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶², 16441, F₄, 28) (dual of [16441, 16279, 29]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(4¹⁶¹, 16439, F₄, 28) (dual of [16439, 16278, 29]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(20) [i] based on
      1. linear OA(4¹⁴⁸, 16384, F₄, 29) (dual of [16384, 16236, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(4¹³, 55, F₄, 6) (dual of [55, 42, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹³, 63, F₄, 6) (dual of [63, 50, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. linear OA(4¹⁶¹, 16440, F₄, 27) (dual of [16440, 16279, 28]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4¹⁶¹Â > V_b^sâˆ’1(kâˆ’1)Â = 253473 623195 315634 881111 504064 189772 532525 068509 905644 918699 106368 238159 232919 363416 735410 177884 [i]
  3. linear OA(4⁰, 1, F₄, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(162−28, 162, large)-Net in Base 4 — Upper bound on s

There is no (134, 162, large)-net in base 4, because

26 times m-reduction [i] would yield (134, 136, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (162−28, 162, s)-Nets in Base 4

(162−28, 162, 1272)-Net over F4 — Constructive and digital

(162−28, 162, 1286)-Net in Base 4 — Constructive

(162−28, 162, 16441)-Net over F4 — Digital

(162−28, 162, large)-Net in Base 4 — Upper bound on s

(162−28, 162, 1272)-Net over F₄ — Constructive and digital

(162−28, 162, 16441)-Net over F₄ — Digital