MinT - Best Known (75−15, 75, s)-Nets in Base 4

Best Known (75−15, 75, s)-Nets in Base 4

(75−15, 75, 1065)-Net over F₄ — Constructive and digital

Digital (60, 75, 1065)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 15, 37)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
    2. digital (4, 11, 20)-net over F₄, using
      - linear code embedding found in a computer search [i]
2. digital (45, 60, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 15, 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]

(75−15, 75, 4217)-Net over F₄ — Digital

Digital (60, 75, 4217)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁵, 4217, F₄, 15) (dual of [4217, 4142, 16]-code), using
- 107 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 27 times 0, 1, 53 times 0) [i] based on linear OA(4⁶⁷, 4102, F₄, 15) (dual of [4102, 4035, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
    1. linear OA(4⁶⁷, 4096, F₄, 15) (dual of [4096, 4029, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [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(4⁰, 6, F₄, 0) (dual of [6, 6, 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]

(75−15, 75, 2608739)-Net in Base 4 — Upper bound on s

There is no (60, 75, 2608740)-net in base 4, because

1 times m-reduction [i] would yield (60, 74, 2608740)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 356 812485 644571 575209 769595 444330 995002 537480 > 4⁷⁴ [i]