MinT - Best Known (80−16, 80, s)-Nets in Base 4

Best Known (80−16, 80, s)-Nets in Base 4

(80−16, 80, 1062)-Net over F₄ — Constructive and digital

Digital (64, 80, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 16, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 8, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
2. digital (48, 64, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 16, 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]

(80−16, 80, 4216)-Net over F₄ — Digital

Digital (64, 80, 4216)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁰, 4216, F₄, 16) (dual of [4216, 4136, 17]-code), using
- 101 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 26 times 0, 1, 52 times 0) [i] based on linear OA(4⁷⁴, 4109, F₄, 16) (dual of [4109, 4035, 17]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [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(4⁶¹, 4096, F₄, 14) (dual of [4096, 4035, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(80−16, 80, 1315729)-Net in Base 4 — Upper bound on s

There is no (64, 80, 1315730)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 461504 024015 958564 069072 621259 119339 823348 408134 > 4⁸⁰ [i]