MinT - Best Known (80−14, 80, s)-Nets in Base 4

Best Known (80−14, 80, s)-Nets in Base 4

(80−14, 80, 2351)-Net over F₄ — Constructive and digital

Digital (66, 80, 2351)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (57, 71, 2341)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷¹, 2341, F₄, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(4⁷¹, 16387, F₄, 14) (dual of [16387, 16316, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷¹, 16391, F₄, 14) (dual of [16391, 16320, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(4⁷¹, 16384, F₄, 14) (dual of [16384, 16313, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(4⁶⁴, 16384, F₄, 13) (dual of [16384, 16320, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 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]

(80−14, 80, 16202)-Net over F₄ — Digital

Digital (66, 80, 16202)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁰, 16202, F₄, 14) (dual of [16202, 16122, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸⁰, 16394, F₄, 14) (dual of [16394, 16314, 15]-code), using
  - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
    1. linear OA(4⁴, 5, F₄, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
      - dual of repetition code with length 5 [i]
    2. linear OA(4⁵, 5, F₄, 5) (dual of [5, 0, 6]-code or 5-arc in PG(4,4)), using
      - complete OA / dual of code with only one code word [i]
    3. linear OA(4⁷¹, 16384, F₄, 14) (dual of [16384, 16313, 15]-code), using
      - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(80−14, 80, large)-Net in Base 4 — Upper bound on s

There is no (66, 80, large)-net in base 4, because

12 times m-reduction [i] would yield (66, 68, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (80−14, 80, s)-Nets in Base 4

(80−14, 80, 2351)-Net over F4 — Constructive and digital

(80−14, 80, 16202)-Net over F4 — Digital

(80−14, 80, large)-Net in Base 4 — Upper bound on s

(80−14, 80, 2351)-Net over F₄ — Constructive and digital

(80−14, 80, 16202)-Net over F₄ — Digital