MinT - Best Known (131−28, 131, s)-Nets in Base 4

Best Known (131−28, 131, s)-Nets in Base 4

(131−28, 131, 1045)-Net over F₄ — Constructive and digital

Digital (103, 131, 1045)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [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]

(131−28, 131, 3581)-Net over F₄ — Digital

Digital (103, 131, 3581)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³¹, 3581, F₄, 28) (dual of [3581, 3450, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³¹, 4118, F₄, 28) (dual of [4118, 3987, 29]-code), using
  - 1 times truncation [i] based on linear OA(4¹³², 4119, F₄, 29) (dual of [4119, 3987, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(24) [i] based on
      1. linear OA(4¹²⁷, 4096, F₄, 29) (dual of [4096, 3969, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(4¹⁰⁹, 4096, F₄, 25) (dual of [4096, 3987, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(131−28, 131, 866746)-Net in Base 4 — Upper bound on s

There is no (103, 131, 866747)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 410720 022441 012409 868653 912082 305183 219250 611921 687417 919775 362665 701391 386204 > 4¹³¹ [i]